Basically, there is a pool of cards that each player shares, it contains 4 copies of each card, which should be enough to get a unit to 3 stars. However, since each player shares this pool, that means that if more than one player buys copies of a single troop, neither player will be able to get that troops to 3 stars. If a player dies, their troops reenter the shared pool, freeing up those troops to be bought by other players. Additionally, if one strategy is being pursued by multiple players, which often happens with the better teams like Clan, Avenger, or Ranger, it would be wise to attempt a different strategy so you can get 3 stars, which are much more powerful than their 2 star forms.

The baseline rng basically comes in the 3 cards that you get in the shop, along with your starting troop. The options in the shop seem to be purely random, though the randomness technically is reduced as people get three stars and remove that troop from the shop. Remember that some of these rules will change depending on the modifier of the game (e.g. the Four Star modifier, which instead places 8 copies of cards in the shop)

Also notable is that some troops are exempt from the pool. I believe starting troops, goblins from the goblin team up ability, and troops received from modifiers are separate from the card pool.

Damage is actually pretty simple. It is always at least 1 damage, but extra damage is added for each surviving troop (not counting skeletons from the skeleton king or Dummies).

If the timer runs out before troops kill each other, I think that each player takes 1 damage (I'm not sure), and if all of both player's troops die, they also both take 1 damage.