To start the answer, I'm going to post the descriptions of Fatigued and Exhausted:
A fatigued character can neither run nor charge and takes a –2 penalty
to Strength and Dexterity. Doing anything that would normally cause
fatigue causes the fatigued character to become exhausted. After 8
hours of complete rest, fatigued characters are no longer fatigued.
An exhausted character moves at half speed, cannot run or charge, and
takes a –6 penalty to Strength and Dexterity. After 1 hour of complete
rest, an exhausted character becomes fatigued. A fatigued character
becomes exhausted by doing something else that would normally cause
1: If you have your familiar do any dex (if they're tiny this is the only one that matters) or str related skill checks, they are hampered (fly, climb, escape artist, swim). This also affects reflex saves, as they are based on your dexterity, so if you get hit by a fireball and your familiar is in your square (to provide the familiar bonuses) they immediately have a lower chance to succeed, because they still use their ability modifiers:
Saving Throws: For each saving throw, use either the familiar's base
save bonus (Fortitude +2, Reflex +2, Will +0) or the master's (as
calculated from all his classes), whichever is better. The familiar
uses its own ability modifiers to saves, and it doesn't share any of
the other bonuses that the master might have on saves.
2: Yes, they are sentient magical beasts with intelligence scores, and as such can be convinced to be on watch. Though if you aren't awake as well to enforce it, the familiar might just go to sleep or something else because its up to the GM to decide how familiars act as they are essentially intimately player related NPCs and there are GMs who would say that you're giving them control of the familiar since your character isn't there to direct it.
3: The only one I'm not sure of based on the rules. The only thing Empathic link says is that you communicate general emotions, not how strong those communications are. As a GM, I'd probably give you a will check or something because otherwise, why wouldn't it being bored or angry at an ant or something jolt you immediately awake?
Bonus: Familiars are not mindless constructs or vermin that you can just give a command to and they obey it no matter what, especially at higher levels. Starting at level 9, a familiar has 10 intelligence which is average human intelligence, and if you tried to tell a dedicated servant to stay up for 3 days straight, he'd probably try his best, but give up once exhaustion set in for the first or second time. And if you regularly mistreat your familiar, it may decide to just leave you even if it means you abandon it and get a new one and it just goes back to being a normal animal.