Rulings

If an exiled card has {X} in its mana cost, X is 0 for the purpose of determining its mana value and you must choose 0 as the value of X when casting it without paying its mana cost.
Paradigm creates a delayed triggered ability that triggers at the beginning of each of your first main phases for the rest of the game. When that ability resolves and the copy is created, you can choose not to cast the copy. In that case, the copy will cease to exist the next time state-based actions are checked. The delayed triggered ability will still trigger again at the beginning of your first main phase on your next turn regardless.
You cast the spells while Improvisation Capstone is resolving and still on the stack. You can't wait to cast them later in the turn. Timing restrictions based on the cards' types are ignored.
Each spell with paradigm has the Lesson subtype. The Lesson subtype appears on instants and sorceries and has no inherent rules meaning. Other effects (such as the learn ability from the original Strixhaven release) may refer to it.
Once a spell with paradigm has resolved, it doesn't matter what happens to the card in exile. Even if the card leaves exile, the delayed triggered ability will still trigger during each of your first main phases, and the copy will still be created.
If you cast a spell "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those to cast the spell.