[RPG] Does Assassinate come into play whenever the Rogue catches the target off guard

Assassinate gives you advantage on attack rolls. So if a spell requires you to make an attack roll than yes, Assassinate can be used in conjunction with that spell.

If it was only meant to be used with weapon attacks than it would specifically say so.

There's one small problem with all this. Here's what Great Weapon Master lets you do:

On your turn, when you score a critical hit with a melee weapon or reduce a creature to 0 hit points with one, you can make one melee weapon attack as a bonus action.

The rules for bonus actions include this all-important clause:

You can take only one bonus action on your turn, so you must choose which bonus action to use when you have more than one available.

So yes, every time you hit while your target is surprised, it will be a critical hit and trigger a bonus action attack from Great Weapon Master. But you can only ever make a single attack that way, because it requires a bonus action, and you can only take one bonus action each turn. And, as @Secesptius points out, if your second attack came from dual-wielding, that required a bonus action, so in that case, you won't be able to get a bonus action attack from Great Weapon Master at all.

For your specific questions:

1. Yes, if you surprised the target, they're surprised until the end of their first turn, so every attack will be on a surprised target.
2. Since they're attacks on surprised targets, each attack will trigger Assassinate.
3. Yes, since they trigger Assasinate, they are critical hits and trigger Great Weapon Master.
4. Yep, as before, they're surprised until the end of their first turn.
5. Therefore any further attacks you make would trigger Assasinate. Unfortunately, as discussed, you can't make any further attacks.
6. You still can't make more attacks, so there's no attacks to loop with.
7. Hiding would also require an action or bonus action, and you still don't have either of those.
8. They would still be surprised regardless of whether you hid.
9. Still out of actions to attack (and thereby Assassinate) with.
10. Same again.