In the case of tripping, IUS allows you to initiate the trip attempt without provoking an AoO. This is because the reason for this AoO is specifically that you're using an unarmed strike for your touch attack. If you're not using an unarmed strike, such as tripping with a Spiked Chain or an Improved Unarmed Strike, you do not generate an AoO.
So far so good. Grappling, however, doesn't specify that the AoO you generate when you try to grab someone is because you're using an unarmed strike. In fact, the AoO is inherent to the grab attempt. IUS doesn't do anything to change this. Improved Grapple is another feat (that requires IUS), which does allow you to avoid the AoO on a grab attempt.
So if you're trying to trip someone with IUS but without Improved Trip, you don't incur an AoO. If you try to grapple someone with IUS but without Improved Grapple, though, you do incur an AoO. Clear?
Prone AoOs: Being prone doesn't change the fact that you're threatening the squares around you, and doesn't stop you from making AoOs. Note that this was different in 3.0 (then, you wouldn't threaten an area when prone, except for the square you were lying in).
Weapon Finesse/Intuitive Attack: These only work on attack rolls, which is the important bit. So while you can make a grapple check as an attack, you're not actually making an attack roll, but an opposed grapple check. Note that Weapon Finesse/Intuitive Attack can be applied to the touch attacks made to initiate a grapple (grab attempt), or to initiate a trip attempt etc. They also work with disarming, since that is an (opposed) attack roll.