Where I will compile various user manual related things:

Assimilation:
Go look at the assimilation manual.

Biogenesis:
I had to un-flexible the flexible traditions in Biogenesis to let you take all 3. When you complete the Biogenesis situation the purity tree will be all purity traditions, cloning all cloning traditions etc...

Auto-Modding:
As of 4.0 Auto modding traits do not stack with each other.
Fixed in 4.0.7, they should now stack. Whether they do or not is core game code and I can't touch it.

Synths
When you take synthetic evolution you get to pick a percentage of your pops and leaders to stay biological. If you pick biological leaders you will get cybernetic/bio authorities. If you pick synthetic leaders you get synthetic authorities. This choice is irreversible.

Your bio pops are handled by letting the situation run its course (turning everything into syths), then a day after the situation finishes an event will run to undo it all. This allows for maximum compatibility with vanilla / other mods, even if its a bit weird.

Multiple Machine Age Ascensions
Do the situation as per normal to get your first tradition. When you complete it there will be an edict to take the others. Obviously nanite and virtual are explicitly opposite playstyles, deal with it how you will.

Multiple Bio ascensions
Do the situation as per normal to get your first, then an edict will appear to let you take the other 2.

Running out of tradition trees
You need a mod to increase them. The mod should also increase the max_tradition_trees variable otherwise the vanilla events will work strangely/not work.

Letting synth empires take machine traditions
It does not, nor do I intend to add the functionality.

Letting Hives take psionic
No, the psionic event system is not designed to work with gesalts. You need to wait until the psionic DLC where I am sure PDX will add them.

Cybernetic Situation turns my species into some random ones
This is a vanilla bug in 4.0. https://forum.paradoxplaza.com/forum/threads/stellaris-native-species-changed-to-a-random-species-during-cybernetic-ascension.1713311/

Advanced Governments
If you are a regular empire who has taken multiple bio/cyber ascensions you get an edict that lets you swap between advanced government types for any of the ascensions you have taken.

AI
The AI is specifically banned from taking multiple ascension paths, or taking multiple of machine age/biogenesis paths.

All Origins
While I have (mostly) origin unlocked ascension paths, I make no guarantees they will all work in a sane way. If you take an ascension that was not designed to work with a specific origin and it acts weird or breaks horribly, that is on you.
Similarly I have not unlocked the Origins that prevent you taking any ascension because your species is already perfect just the way it is.