How does class progression for unlockable passives and skills/combat arts work in Fire Emblem 3 houses?

Solution 1:

The Class Mastery system works as follows:
In addition to each character's overall level, they have an additional "experience bar" for each class they are taught. We'll call this class mastery, or CM.
The character's CM goes up by 1 each time they do something that would trigger the battle animation.
For example, attacking, getting attacked, or healing. But not using a Rally ability.
This can be increased to go up 2 each time by spending your renown on the Saint Cethleann Statue. 2 Appears to be the max you can get. Through my testing (with the Saint Cethleann Statue buff), this seems to go up by 2 all the time, even if your sword unit is attacked by a bow and you don't hit them back, or if you hit them 4 times with fist weapons, or if you crit, it always goes up by 2 (1 without the Saint Cethleann Statue buff).

Each level of class (Starting, Beginner, Intermediate, etc.) has a max amount of CM that can be earned for that class, and once that max CM is obtained, the character will learn something extra (usually a new ability) for the rest of the game, even if they change off that class*
The wiki has a good list of what characters obtain for mastering their classes under the "Mastery" column.
I have not yet found information online regarding how much CM is needed, but in my own experience playing through twice (once Normal, once Hard), i know the following ('~' means its around this number, as I remember)
"Starter" (Commoner, Noble): 20 CM
Beginner: 50 CM
Intermediate: 100 CM
Advanced: ~150 CM
Master ~200 CM

For your other two specific questions:
Not seeing the bar might be due to you turning off combat animations, and

If both Lysithea and Marianne have mastered the monk class, they should both have Draw Back and Magic+2.
One thing to check would be their Abilities screen, and their Combat Arts screen.
Units can only have 5 abilities active at once, and only 3 combat arts,
so if Marianne already had 3 combat arts (most likely lance combat arts), her Draw Back would have automatically been stored since it could not be equipped.
If you go into Inventory > select Combat Arts > select Marianne, you should see what she has equipped, and what she has learned, and be able to swap between them. Same goes for Inventory > select Abilities.

I've found the best way to see how close you are to mastering a class it to wait until a Sunday, before you select Explore/Battle etc., select Certifications.
From there, select your character you want to see, then select the level of class you are looking at.
For example if your Marianne is currently a Priest, and you want to see how close she is to mastery, select Intermediate, if you want to see how far she got as a Monk, select Beginner. Once you have selected the level of class, all the classes of that level will be available to view to take a test and get certified, if you select a class for which you are already certified, in our example the monk class, it should either show a little yellow star in a purple circle, meaning you mastered it, or it will say X/50, showing how close you are to the 50 CM points needed to master that class.

*Some class mastery rewards are combat arts that can only be used while that character is on the field as that class, such as the sniper's "Hunter's Volley" or assassin's "Assassinate", the assassin's "Lethality" ability still works on other classes however. I learned this the hard way :(

Solution 2:

The class mastery definitely could be better telegraphed, but the main places to see it are in combat when gaining XP (yellow bar means maxed) and in the one page stats screen where it will be shown after all the other skill ranks with a sprite icon of the character as that class (the one with all the skills as icons in the bottom right rather than as a list on the right).

You can also tell if the class has a little badge in the reclass/certification menu, if it's not there that unit has not yet mastered it's current class.

As far as I am aware, units earn 1EXP towards their class mastery each time they participate in combat, which can be increased to 2 from a statue upgrade. I've also heard that the Knowledge gem helps though I have not verified that myself.