Some questions.

[Malek]

Can someone remind me:

1)Do prone characters still can make Attacks of Opportunity?

2)If I sucessfully trip someone while having "Improved Trip" feat can I use the extra attack from omproved trip to start a disarm?

[Thomas]

1.

Prone: The character is on the ground. An attacker who is prone has a –4 penalty on melee attack rolls and cannot use a ranged weapon (except for a crossbow). A defender who is prone gains a +4 bonus to Armor Class against ranged attacks, but takes a –4 penalty to AC against melee attacks.
Standing up is a move-equivalent action that provokes an attack of opportunity.

Threatened Squares: You threaten all squares into which you can make a melee attack, even when it is not your action.

You can make melee attacks while prone, ergo you threaten, ergo you can take attacks of opportunity. There's no exception mentioned.

2.
Yes, you can. You can turn Cleave attacks, attacks of opportunity, etc. into disarm, trip, or sunder attempts.

[Person_Man]

It's also worth mentioning that once a person is Prone, they cannot be Tripped again until they are standing up. When they go to start standing up and provoke an AoO, they still count as Prone until after the AoO is resolved. This prevents serious Trip build abuse.