TL;R - It's not the total number of potions, it's your overdose threshold - whatever you raised that to.
I've only partially spec'd into alchemy, but even with that have total tox max 204, and can drink 7 potions and still climb ladders. The 8th is what overdoses me.
Decoctions though are so much stronger, can only drink 2 decoctions + 2 potions, so again, total number will depend on your build.
Can't run while burdened, so while don't know if this one is also intended design or bug - leaning towards intended design. Seems reasonable consequence to being overdosed.
Tested it myself and the trigger is not X number of potions, but whatever is your overdose threshold - so if you took the alchemy spec for acquired tolerance and heightened tolerance - your number of potions able to use concurrently can be quite higher than just 3. With only partial spec into alchemy, I can drink 7 normal potions and still function just fine. Or some mix of stronger decoctions + normal potions, long as I am under my overdose threshold.
Yes - likely overdose why you couldnt climb on boats. Can't say for sure aren't other reasons why climb may not work, but for sure if overdosed, it won't work.
Probably no - not a glitch, but likely intended design. I say probably and likely because can't read devs minds, but seems logical since other movement functions are disabled under adverse conditions (e.g. No run while burdened)
Yes and No - clearing tox would resolve get back climb ability, but technically you dont need to full clear toxicity, just reduce your tox threshold below the overdose threshold