I've leveled up conjuration 10 times, and I also leveled up alchemy and mysticism quite a few times as well, but it still only shows +2 next to intelligence when I level up. Isn't it suppose to be +5 or at least +3?

Nope. For that amount of leveling of intelligence based skills, +2 is correct.

I thought leveling a skill 10+ gives the attribute it's assigned to +5 when you level up. And after I leveled up, I went to sleep immediately and I leveled up again, and then I went to sleep again and leveled up without doing anything. I'm confused...

The game continues to track increases to your major skills between achieving the level up and actually sleeping to gain the level. These increases are counted toward the next level so if you increase major skills 20 times then you will level up twice in a row.

You may have increased conjury 10 times but I am assuming it is a major and that you also increased other majors at the same time. and the 10th major skill up only counted four increases to conjury.

The final 6 increase would count to the next level giving a +2 for the first level up and +3 for the second.

When your character is ready to level up (10 points of the Major skill were raised) then any skill gains you make after that, minor or major, wont apply till your next level up. You'd have to raise attribute skill/s by 10 points in between level gains to get a +5. So if your character is at level 1 and he raised his conjuration by 10 points before reaching level 2 then he'd get +5 to intelligence.