Do health potions share a cooldown with stat potions (agility, strength, intellect, etc.)?
Potions and elixirs are on different CDs. I think Healing and Mana are Potions, and Stat buffs are Elixirs.
I might be wrong. 
No. Health potions are on a separate cd. Mana potions do share a cd with those other dps/utlity pots, though.
Thanks for the replies. Looks like I’m switching to Alchemy.
It’s always a good idea to have a character with alchemy. Such a useful profession.
Random side opinion - Why did they increase the cooldown on Potions from 2mins to 5mins?
I mostly want those movement speed potions to return to 2mins 
I think their explanation had something to do with people feeling like pre-potting in raids was an absolute requirement. Now potions are something you use once or twice per fight to line up with your cds, rather than always having to use one before the pull.
It has been a while since I alchemied. Is the skill that procs brewing extra flasks still a thing?
Unfortunately not. But flask mats aren’t nearly as much of a pain to farm as they were previously. Korthia and ZM are packed with Nightshade.

Seems much more useful this go-round than Engineering. I do love my bombs though. 
As far as I know ,no.
If you don’t plan on using professions it’s a great passive option. It gives you an extra hour duration on flasks!
Yes all potions are on the same GCD.
It’s why first aid being removed out of retail was silly.
The best Alternative is Warlock Healthsone if you’re planning to use battle potions.
Agility would be a Battle Elixir. So you could use a guardian elixir if you wish. Does go away if you die.
A flask takes the role of battle and guardian elixir and battle elixir and persist through death.
Yeah, I figured there was an explanation for the utility increase from 2mins to 5mins.
I still wish the psychopomps speed potions would be reduced to 2mins, seeing as it isn’t the same tier of potion as the stat potions.