What is the minimal seat post depth? This may depend on the seat post and the fixture, but how deep should my seat post be into the frame? I have a bike that came with a post that is pretty long and I cut an inch off the bottom of the post a couple months back so that I could sit lower, now I'm considering cutting two to 4.5 more inches off so I can get further down on the bike during some technical riding and so that I can pump deeper. How far into the bike does the seat post need to go before stability is compromised? Are there any other risks of having your seat post not too deep?
Most seat posts have a "minimum" insert of 3 to 4 inches.  But this varies with the material and the thickness of the post, and the weight of the rider means the minimum might still be too little.

As a tall rider I frequently have my seatposts up to maximum, and have bent several over time, and have fractured one frame.  Now I always buy a 450 mm or 500 mm post and almost always reinforce it with an insert tube up the middle.  I will not ride a seat post without 5" of frame insertion.

Since you sound like a MTB kind of rider, consider a dropper post rather than chopping ruthlessly through your seat poles with a hacksaw.  These are intended to go at a good height for normal efficient riding, but you can turn the interlock and they slide down ready for low-seat technical riding.  I understand you can do it one-handed while rolling too.

Once you're done you can raise the seat again without stopping.  Downside is they probably cost ten times more than a normal seatpost.