Does the bishop stats really matter?

Solution 1:

As far as I'm aware, the tax income and levy size you get from any vassal are based on their holdings, and modified by your realm's taxation/levy laws and the vassal's opinion of you (for taxes at least - I'm unsure whether opinion influences levy size). Catholic Bishops have the added complication of paying their tax to either you or the Pope, depending on who they have the higher opinion of. I don't think a vassal's statistics influence any of these things.

Pretty sure any male of your court can be appointed to the role of Court Chaplain - on appointment the character's portrait will change to show them wearing religious garb.

You're right that making unlanded son's Bishops counts as giving them land. Be careful with this though, as once made a Bishop your son is not eligible to inherit other titles from you in the usual way. If your main heir has an "unfortunate accident" you may find yourself suddenly lacking an eligible a successor!

For realms smaller than Kingdom, any lord can appoint a successor for any Bishop that is his direct vassal. If no appointment is made, the new Bishop is appointed internally by the Church (or Pope). Larger Catholic realms must opt for one of either Free or Papal Investiture - Free operates as before but angers the Pope, while Papal gives the Pope sole control of Bishop appointments in your Kingdom and grants you the Pope's favour (useful when seeking to excommunicate rivals, divorce wives and so on).

Solution 2:

Martial matters but Stewardship does not matter (much)

When you hover over the red Levy bar below the picture of their church on the county screen, you will see a percentual malus or bonus for "Owner Martial Skill" which depends on the martial skill of the bishop/baron/burgher of the holding, not their boss. So you should make sure that your lowest-rank vassals have good martial skill.

But stewardship of bishops (and other vassals on that level) has no direct benefit for you. I happen to have two completely unupgraded churches in my realm, both owned by bishops with 100 opinion. But one has a stewardship of 10(20) and the other a stewardship of 0(0) (my imbecile brother who had to be removed from inheritance). Both have the exact same Base Tax of 8.00 gold. The actual Tax value which arrives at the liege is modified by the stewardship of the liege, not of the vassal: The imbecile is below a count, and that count gets slightly more out of him than I get out of my guy, because the count has slightly better stewardship than me.

However, better stewardship means that they make more money for themselves, which they might then invest into upgrades for their holding. Unfortunately higher-level upgrades are far outside of the financial capabilities of a bishop, even one with godlike stewardship.

Other stats don't matter much either

• Diplomacy might help them a bit to stay out of trouble with other people, but that is rarely a problem for such low-level vassals.
• Intrigue of your vassals should better be low, so they don't start murdering people you would prefer to keep alive or get other stupid ideas. Low intrigue also makes it easier for you to get them out of the way through murder or imprisonment, should it become necessary.
• Learning only matters for tech growth and their personal piety. The first only matters for counts and above and you couldn't care less about the latter.

So the ideal stats for a baron-level vassal is a stupid brute with awesome martial skill and absolutely no other talents.

Regarding your court chaplain question:

You can't make a noble or burgher your court chaplain, but unlanded courtiers are just as eligible as bishops. So when you want to keep your option open to make that high-learning guy your chaplain one day, just remember not to accidentally give him a city or castle.

Screenshot Proof

Edit: Here is a proof which uses two other bishops I control directly which shows that their stewardship doesn't matter for me. I present to you my trusted bishops Ualan and Orthanach (the latter also happens to be my brother, but that shouldn't matter). As you can see, Ualan has far better stewardship:

They own the exact same church, but the one owned by Ualan has a higher income even though his stewardship is lower. I believe this is because his church is in my capital, so he enjoys a better bonus due to church infrastructure technology.

According to their demesne screen, Ualan gets more out of his church than Orthanach, despite his worse tech bonus. That's where his stewardship pays off. But, they both pay almost the exact same tax to me (difference below 1% - likely due to rounding). For the record, my church vassal tax is 45% (medium):

However, the tax I receive from either is more than what they pay to me, but still almost the same. I presume that this magical money-growth comes from my stewardship bonus, my tech bonus or both:

Solution 3:

To at least some small degree, all of the stats of a vassal bishop (and vassals generally) matter or can matter, but martial and stewardship are the only ones that have a direct and immediate effect. The direct effect of the martial attribute on the size of the levies your vassals will raise for you was addressed in Philipp's answer so I won't address it here. There is also a similar bonus to the taxes you receive that are affect by the vassal's stewardship stat and not by your own.

Stewardship Matters

State stewardship, which is the sum of the ruler's stewardship attribute, half his wife's, and all of his steward's, provides a 2% bonus to that ruler's demesne income for every point over 5. Note that bishops can have wives, and while bishops don't have councillors, all baron rank rulers act as their own councillor. A vassal's total income is his demesne income plus all the taxes he receives from his vassals. A bishop can't have any vassals so his demesne income will be his total income. The total income is the multiplied by the vassal's taxation percentage, which is determined by the type of title the vassal holds (feudal, mayor or church) and the current taxation law for that kind of title. The vassal passes on the taxes to his liege and keeps the rest. A bishop can choose to send the taxes to a Pope instead, one that is both his religious leader and that he likes better than his liege.

Proof

As proof I offer the following example from the end of my game playing as the de Normandie dynasty. I've chosen a prince-bishop as an example as the numbers involved are bigger and because it's a more complicated case that better represents how vassals are taxed generally.

The tool tip shows that Prince-Bishop Christopher's demesne income has total base value of 25.4 before being modified by his state stewardship. The percentage shown appears to be a bug, the actual percentage used is 30%, which is what we would expect given his state stewardship of 20. After the stewardship bonus is applied he receives a total of 33.10 from his own holdings. Add in 49.44 in taxes collected from own his vassals he has a total income of 82.54. Of that he pays to 28.88 to his liege in taxes. That's the expected taxation rate of 35%, given that his liege's realm has the Minimal Church Tax law in effect. If the vassal's stewardship wasn't a factor in the taxes the liege receives we would expect that only (25.4 + 49.44) * 35% = 26.19 would have to be paid.

The same is true for an ordinary bishop:

The amount paid the bishop's liege is 35% of his total income, which since he has no vassals is also his demesne income. If the bishop's stewardship didn't apply to taxes then he would be paying his liege only 25.6 * 35% = 8.96.

We can also see that the liege's stewardship attribute doesn't affect the taxes he collects from either vassal:

There's a tiny difference between if you add up all the vassal bishop's individual taxes and the total church tax due to rounding, but no where near what it would be if the liege's stewardship was taken into account. Like with his vassals the liege's state stewardship only affects his demesne income, not his income from taxes.

Indirect Effects

Your bishop's traits and attributes can have other indirect effects, many if of which won't matter much if you're an emperor, and some that could matter a great deal if you're a count. Promoting a landless character to bishop means that they now become players in the game, for good or bad. Even if they're beneath your notice their actions will either tend to benefit you or harm you. The success of these actions, along with whether they're attempt or not, will all in some degree be affected by one or more of their attributes and traits.

Military

Promoting someone from your court to bishop, in realms with a least Limited Crown Authority, will allow you to assign them as commanders in your armies. Even if the holding is poor and undeveloped, making the martial levy bonus insignificant, you might still want to nominate someone with a high military attribute so he can participate in your battles. Such a holding might be ideal for an ambitious family member in your court who's martial abilities would otherwise go to waste because you don't want to risk giving them a better title.

Stewardship

Since a vassal's stewardship attribute affects how much money ends up in his own pockets, it affect how often your vassal can afford to upgrading his own holdings. This increases both taxes and levies the holding generate for the vassal, which in turn increases the taxes and levies you receive from the vassal. Effectively your vassal's stewardship provides a additional indirect bonus to your income and levies, one that can potentially snowball into a huge benefit in the long term.

Diplomacy & Learning

If you're playing with the Sons of Abraham DLC then diplomacy and learning will have direct and indirect effects on your vassals chances on becoming a Cardinal, and if a Cardinal being elected Pope. Numerous traits also effect the likelihood of gaining either of these positions, so if you have this DLC you'll want to consider your candidates stats fairly carefully when nominating bishops.

Traits

Most traits only matter for the effect they have the character's attributes, but there are a number you might want to consider when appointing a bishop. The Greedy trait gives a 10% bonus to demesne income and so increases he taxes you collect in the same way the vassal's stewardship attribute does. The Content, Envious and Ambitious traits will all affect your new vassal's opinion of you. In particular watch out for the Ambitious trait, as it might not factor in a character's opinion of you while he's just a courtier. Traits that change a character's health affect how long he lives, while the military traits will affect how well he performs as a commander in your armies.

Granting Bishoprics

In addition to nominating bishops, you will probably from time get the opportunity grant a church holding directly from your demesne. This opportunity usually comes about after creating a new holding or gaining some though a holy war. In this case attributes can make a big difference because you can grant the title to anyone in your realm, not just to someone in your court. Say for instance a duke in your kingdom has an amazing spymaster with 25 intrigue. Assuming that the spymaster is landless (and not his wife) you can grant the spymaster a church holding. The spymaster will now become your vassal, giving up his position in the duke's council and now eligible to hold a position in yours. If you're rich enough it might be worth it to create a church (or other) holding just for this purpose.