What causes Elizabeth to give you the different kinds of loot?
I noticed that Elizabeth will occasionally comment that she doesn't have anything to give yet. How often will she get loot to give to you and how does it work? Is the amount of loot she'll be giving you fixed or decided by certain factors?
I noticed that Elizabeth will give Silver Eagles at first, then later, ammo, health, and salts (not sure if it's in that order). Are there specific points in the game where she'll start giving a different type of loot? If yes, at what points in the game will she start giving each different loot type? (I didn't notice at what point of the game she starts giving ammo, health or salts.)
You could accept loot from Elizabeth while in the middle of fighting. How does this affect gameplay? Will I take damage while I'm catching loot from Elizabeth?
Elizabeth will give you things based on a few factors:
Your progress in the game. She will start throwing you things after you have a conversation with her about them. For instance, she starts throwing you money after Battleship Bay, when you encounter a vending machine. She'll mention that she's going to keep an eye out for money. Later, as you're heading towards Hall of Heroes, she will start throwing you weapons.
There's always a dialog the two of you have before she gains a new "ability" - she'll say something like "Those Vigors seem to be quite powerful, I'll keep you stocked in Salts" before she starts throwing you Salts in combat. I believe the unlock order is:
(With thanks to hammar for jogging my memory about the precise order and locations on these)
The current situation and your combat strategy. If you're low on health, she'll toss a health kit. If you're low on ammo, she'll toss you a fully loaded weapon (always the same one I was using when she called out to me) and some extra ammo.
When she says she has something, she is at that point locked in to whatever she's holding. You can drop/change weapons, and she'll still throw you the one she's got - you'll drop your current weapon if you don't have one of whatever she's tossing. In my experience, health takes precedence, and then whatever combat action I was taking when she first started noticing I was low.
The difficulty and how many times in the current engagement she's helped you. For money, there's almost certainly triggers in the environment that count. You can leave the game idle, and she'll not throw you any money. However, it seemed like whenever I was near a shopping area with vending machines, she would toss me some extra money.
Usually she will notify you that she's "looking" for things when your ammo, salt, or health is about half, and then later she'll throw you something. This can be a good indication that it's time to change tactics - she'll reliably warn you that you're running low on ammo or salt, for instance.
In combat, you will only get so many items from her before she'll declare she's all out. When she declares she's out, you're out of luck. I've been down to my last clip in both weapons, a sliver of health, and no salts to speak of, and she'll just cower in the corner. At that point, it's skyhook time.
I've found that at harder difficulties she can't find as much stuff.
Some other notes about Elizabeth's game mechanics:
Elizabeth can also open rifts to give you a combat advantage (this starts in the Hall of Heroes), but that's another topic... :)
As a side note, she will also hover over any lockpicks in the general area and let you know there are lockpicks. Handy. :D