How to prevent the kings from changing their succession law within the empire

It's hard to prove a negative, but as of version 1.1 it doesn't seem so.

I did a text search on the game's script files which describe all the interactions, events, decisions etc. in the game (\Crusader Kings III\game\common\) for the string remove_trait = drunkard and received 0 results. I then did a search just for drunkard to see if there is maybe a different syntax for removing a trait. I found a lot of results which check for that trait's existence or add it, but none which seem to remove it.

So apparently the one enemy you can not defeat in CKIII is alcohol.

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.