Is there a way to remove all items from squad members, or exchange items between squad members?

No, this is a relatively common interface complaint - I've seen it several times in various forums. If you want to find a particular item, you've got to dig through all your soldiers to find it.

From experience, wounded soldiers return their items to the central pool while they are on medical leave, and then I believe they'll try to reequip when they come back to active duty. However, soldiers on Psi training do not do the same. You'll have to go to the barracks and manually unequip their stuff. (This is a major pain if you don't realize this before you've accepted a mission, although you can still back out and go to the barracks, and then back to mission control when finished)

For certain "cheap" items like medkits, I suggest overproducing and just hand them out like candy. With more expensive stuff like high-end weapons and armor, I'll tend to completely strip a soldier (including resetting their equipment back to the default grenade) if I'm taking them out of the active rotation - that way, I can tell at a glance by their weapon and armor in relation to other soldiers whether or not they're combat-ready and/or are potentially holding a key item.

You may like to note that with the Enemy Within expansion come several interface enhancements. Notably for this question, there's a "Make Items Available" button that removes all items from everyone who isn't in the current squad. The pre-mission squad view also shows all the items in everyone's pockets so you don't have to try to remember that.

The expansion is quite expensive, but I used isthereanydeal to watch for when it was on sale.