What can you trade?
Can I trade enemy drops like gel cores? How about guns, upgrades, and trinkets? is there a limit to what I can trade?

All items that are not bound to you can be traded. You can also trade Crystal Energy.
Materials (enemy drops) and recipes can be openly traded. Gear can be traded provided you haven't equipped it/have it bound to you. You can check to see if it is bound to by scrolling over the item in question, and it should say right under the stats.
Upgrades, I haven't actually tried, but I believe they are bound to your character upon purchasing. Trinkets fall under the same category as gear, and can be traded provided they haven't been used before.
That's a really good point: do upgrades automatically bind to the player as they acquire them? Could I trade, say, a Mist Energy Recharge Tank?

Most special items tend not to be trade-able. Though I've never had a Recharge Tank to try it with.
Don't you get a recharge tank when you pass through the tutorial and move on to Haven? I'm just wondering if someone could potentially trade that tank to someone else for crowns or items.

I, too, haven't ever gotten one (started way before they added those), but someone in game had said that they could not trade it.
I actually just tested it by attempting to attach the recharge tank to an in-game letter. It said that it was untradeable. Guess that takes care of that.
You can trade anything in unlimited quantities except Bound equipment, Boss Tokens and Minerals.