Could my monk use a rope or chain as an improvised whip? If so would I be able to trip and disarm without the AoO and with reach when using the Catch Off Guard feat?
Catch Off-Guard (Combat)
Foes are surprised by your skilled use of unorthodox and improvised weapons.
Benefit: You do not suffer any penalties for using an improvised melee weapon. Unarmed opponents are flat-footed against any attacks you make with an improvised melee weapon.
If I am using the Whip Mastery feat from UC do I avoid AoOs when I trip and disarm?
I am thinking that I will need the Improved Disarm and Improved Trip feats to avoid AoOs no matter what...
Using the whip already takes up a ton of feats: Exotic Weapon Proficiency, Weapon Focus, Whip Mastery feats, and I still can't use it to disarm and trip without AoOs???
reach does not come without a price
Could my monk use a rope or chain as an improvised whip? If so would I be able to trip and disarm without the AoO and with reach when using the Catch Off Guard feat?
Not normally, no. An improvised weapon is just an improvised weapon, you don't get extra abilities by calling it an improvised whip. There is an Equipment Trick in Pathfinder Society Field Guide to use a piece of rope as a whip or spiked chain, but it requires proficiency in the weapon and an extra feat for Equipment Trick.
It should be noted, however, that you can actually disarm or trip with any weapon. You simply don't get the +2 on disarm attempts, or the ability to drop your weapon on a failed trip, that are imparted by those abilities.
If I am using the Whip Mastery feat from UC do I avoid AoOs when I trip and disarm?
Nope.
I am thinking that I will need the Improved Disarm and Improved Trip feats to avoid AoOs no matter what...
Yep. Unless you are out of reach, of course.
Using the whip already takes up a ton of feats: Exotic Weapon Proficiency, Weapon Focus, Whip Mastery feats, and I still can't use it to disarm and trip without AoOs???
Pretty much. The weapon has many advantages and disadvantages, and it requires a bucket load of feats to really trick it out. I'd recommend considering just staying out of reach to avoid AoO, and concentrating on one aspect (Whip Mastery, Trips, etc) instead of trying to do everything.