Lots to unpack here! Basically, there are a lot of quite precise ideas that are good to draw out and consider separately.
Starting with your question 3, a structure satisfies a formula ‘Fx’ if the interpretation of the object ‘x’ in the structure - say, u - is contained in the interpretation of the predicate ‘F’. Tarski’s structures interpret predicates using an Extension, which is broadly speaking “the set of things that we interpret the predicate ‘F’ to correctly apply to”.
So, think about what the universal quantifier does - It says everything is F. If we interpret this relative to a structure, we’re aiming at everything in our domain being in the interpretation of F.
By starting from models of simpler formulae, we set our Satisfaction notion up Inductively, and our concept of satisfying quantified formulae is a general version of satisfying elementary formulae.
This inductive definition is what gets us a Compositional theory of Truth, meeting your question 2. The reason we can work out the schema for every one of our candidate “true” sentences is that we’ve defined our satisfaction relations over the first order predicate calculus, using each of its logical compounds to show how the truth of the parts of a compound sentence relates to its truth as a whole.
Interestingly, though, in Tarski’s theory, we don’t get a model of a language that itself contains Truth as a proper part. Doing so in the base theory would result in a circular definition! Instead, you get what is sometimes understood as a Hierarchy of truth predicates; you can define a base language and interpretation, then a compositional first order Truth theory over that, adding a new truth predicate to create a second language and a second-order compositional Truth theory, and so on.
If truth never applies to a language featuring that same truth itself then what grounds do we have calling a predicate a “Truth” predicate? Well, as per your question 1, the point about something being a truth predicate is that it takes a sentence and returns a condition in our interpreting language that we can follow and thereby determine. And it does so in this very transparent way - ask what the sentence says, and say the sentence is true if that thing it says is the case!
Tarski argued that this “material adequacy” condition was the key factor in reasonably calling something an effective Truth definition. His compositional definition meets this requirement, even though it is reasonably conservative about the kinds of statements in its language one can affirm to be True (I.e. in first order predicate calculus without a prior Truth predicate).
Not all theories of Truth meet requirements 1 and 2 - Languages admitting the Liar paradox being a good example! But Tarski’s view set the groundwork on which basic models of predicate languages could be formulated mathematically, which has done a lot of work in logic and computation.