Final Fantasy XIV - How does co-op questing work?

First, you should know that you can use the game's party function while on the free trial, but you will need to get another player who isn't playing on a trial account to form the party for you. Once the party is formed they can leave. You may want to ask a mentor in the Novice Network to help you form a party.

Now on to your questions.

1. Quests are generally designed to be done solo. Many can be done in a party, but you will often be forced to complete solo instances which will require you to disband the party. You will also often find that objectives must be completed multiple times for each party member.
2. The party requirements only apply to instanced content such as dungeons, trials and raids, and only if you are using the duty finder for matching. If you go in as a pre-made party or use the "undersized party" option the party composition requirements are lifted. However, since you are playing in a group of 3, and the smallest party size for instanced content is 4, you will need to stick to the standard composition of 1 tank, 1 healer and 2 DPS unless you use the undersized party option (which will remove any experience rewards, so I don't recommend it).
3. There is (generally) no quest progression sharing. Each person must complete quests individually. For quests that require you to kill a certain number of enemies (that exist in the world regardless of quest progression), the credit is given to everyone in the party, so progress can be shared in that sense, but that's more of an exception to the rule. If two people are in a party doing the same quest and the quest requires you to spawn and kill an enemy, it will need to be done twice.
4. Minor drops such as crafting materials or potions will fall to a random party member immediately. Major drops such as equipment will go into the party's loot pool and can be lotted on using the standard need or greed system used by most MMOs.