Can you drink multiple different potions and have them all be in affect at the same time? Or will drinking another potion simply remove and overwrite the affect of the previous one? For instance, can I drink a +stamina regen, a Trueshot, a +light armor bonus, etc... and have them all in affect at the same time?

Yes, some potion effects stack and add.

What I am not sure about is the timing. If the effects timers are frozen while the inventory window is open or not. In other words, if it takes you 30 seconds to take all the potions, do you lose the effects of the first potion if it is an effect for 30 seconds, or does the timer start when the window is closed.

I believe so, but I could be wrong. Don't think you can stack like potions though, like 30% frost resistance and 40% frost resistance to get 70%.

They stack if it's different potions. For example, a Philter of Health and an Elixir of Health will stack, but if you just drink two Philters the duration will just reset.

Also, timer starts when window closes, since the game is paused while you are in the menus.

Good side note: Timer keeps going while at crafting stations. If you drink a fortify enchanting potion, you still only have the duration of the potion to do your enchanting with the benifits of the potion.

That is why I was wondering about the timer, as when using the Enchanter or Alchemy Bench, you only have so much time.

And it is important to stress that it needs to be different potions that stack. You can stack philter of strength, fortify strength, and a bunch of other strength potions as long as they are of different names. That will help a lot if you are worried about being over-encumbered.

Does the timer stop if you rename your enchanted items... On an xbox it can take a while!

I don't think so. I think Bethesda kind of screwed up in this design - I realize the intent was to avoid the situation in Morrowind, where, e.g. you buff your intellgence, luck, etc with high-power, 3 or 4 second spells, then open the character sheet to begin enchanting, and can enchant as many items as you want with the buffs.

However, time-based limitations svck when your user interface svcks as badly as Skyrims does. You click on things and you get the wrong item or enchantment selected then have to go back and re-select, you want to take time to re-name an item, etc. I think the "right" way to do it would have been for enchantment potions to be effective for a single enchantment attempt. That is, "Your enchant skill is increased by 20 points for the next item you enchant". Either that, or allow the user to actually drink enchant potions from within the enchanter UI - e.g. you select your item, enchantment, and soul gem, rename the item, click the button to drink a potion, be selected with a list of potions containing +enchant effect, select the potion you wish to drink, drink it, then finish the enchantment.

However, it's clear that Bethesda didn't put sufficient thought into the interaction between their clunky, unresponsive, inaccurate user interface, and timed potions.

The timer is supposed to stop when you are in the name box. Whether it does or not is another question. I have not used an Alchemy Potion in this game, so I don't know.

Timer does not stop but the effect does not change if timed out as long as you have selected item, soulgem and effect. So you can take your time renaming.
But if you try to change item, gem or effect after 30 seconds you will lose the potion effect.

Regarding potions and stacking.
Can you stack enchanting and smithing potions if you add other effects to get different names? Worth to try out.