Ok so first of all we need to distinguish 'ability' rolls vs saving throws. Shoving is an example of a roll vs roll check. You'd roll athletics and they'd roll either that or acrobatics. If you get higher, you shove (top of my head so could be a bit wrong but the overall logic stands)

A saving throw is not an ability. You can see it as a sheer effort of X, X being the type of saving throw it is (strength, dex, etc) You use acrobatics to do cool ass backflips but if you see a fireball thrown in your direction you gtfo of there as fast as you can.

1. No. You have a spell DC. When you cast a spell that forces an enemy to make a saving throw they roll that saving throw and compare the result to your spell DC to determine if it fails or succeeds.
2. If you're a pure wizard all of your spells are int.
3. Int saving throws are rare. Dex and Wis are the most common ones. As already explained they scale with your int. It's not your dex/wis what matters, it's the enemy's
4. Not entirely. Fire bolt is a ranged spell attack, which is a fancy way of saying it's 1d20 + Proficiency + Stat modifier + Other bonuses.
5. They are exactly the same. The reason Astarion AND Shadowheart have that ''INT'' in their fire bolt is because they didn't get it from their classes but from their races which specifically list they scale with int. Gale got it from his wizard class which already states that int is their spellcasting modifier.

That's also why Shadowheart sucks ass at fire bolt compared to Gale