Partition vs High Partition

First of all, your titles are more than just counties. Every duchy, kingdom and empire is also a title. If you hold 1 empire, 2 kingdoms, 2 duchies and 10 counties, that's 14 titles, so your heir under high partition would inherit half of that, if he gets the empire, kingdoms, one duchy and 3 counties.

The only difference between high partition and normal partition is, that your heir gets at least 50% of your titles, while under partition, the heir only gets your primary title down to the realm capital (i.e. your capital county, the duchy it is in, the knigdom that is in and the empire that is in) guaranteed.

Yes, it matters, what your children already hold. A child is only eligbile to inherit, if they don't already hold a title of the same or higher tier. Only when every child has at least 1 title of that tier, they become eligible again. As such it's a good idea to not give your primary heir any titles whatsoever, as that increases their gains during succession. If you were to give each of your children a duchy or kingdom for example, while your heir has nothing, you'd only lose 1-3 counties on succession. At least, that's the method I used to get almost all my titles to my heir during high partition.

Another method you can employ is elective. For kingdoms, elective is difficult, as you have to handle a lot of vassals usually, but for Duchies, you hold only yourself anyway(you really don't want direct count tier vassals in an empire), employing elective is a guarantee to get the duchy to your primary heir. With the duchy in hand, you can easily make a claim on the county and revoke it from the child who gains it after the succession without repercussions within a year.

After searching some more, I can across this thread which is extremely dense, but seems to shed some light on the subject:

https://forum.paradoxplaza.com/forum/threads/succession-laws-explained.1419049/

I think the key points for high partition are:

Primary Heir will inherit the lion’s share (as much as possible but not more than 50%)

So it's not a straight 50%, it's a maximum of 50%. Also:

Each subsequent lower tier Title follows the same rule as the previous step where the Primary Heir will get the lion’s share of the remaining Titles of that tier

So the rule is applied to each tier separately but is complicated by the fact the lower level titles that are de jure part of a higher level title that has already been inherited by a child will go with that child and be taken off the table for this round.

And on the subject of children:

Note 1: If an Eligible Child already held Title(s) of that tier when the Succession happens, that Title will also be included in the count to determine eligibility for every Round.​

So my player heir with two kingdoms will automatically get my primary kingdom title (because it contains the capital) but would be ineligible for any other kingdom titles until every other heir has at least three kingdoms.

The one thing that continues to puzzle me is that my ruler has his capital in the duchy of Magadha (in India) and owns all four counties therein, the duchy title, the kingdom title (Bihar) and the empire title (India). But it looks like one of my non-player heirs is set to inherit the county of Gaya which is de jure part of the duchy of Magadha. My reading of the rules is that that shouldn't be possible?