Having been around Armor and Cavalry for a portion of my career, I can help a little. I'll take on your questions one at a time...
Can a vehicle like these MRAP’s survive in an urban area without supporting infantry?
The answer is : briefly. The routine is, the vehicles would punch into the target area (say a city block or mall), the troops would dismount and form a perimeter, the task gets done, then they remount, pop smoke and head back out.
The transit into and out of the area is the scary part, where the troops are tucked inside the vehicles for the ride (as they say in the Cavalry, "Death Before Dismount"). The vehicles, of course, have gunners up and scanning for threats, but the passengers are hunkered down. The MRAP's allow for gun ports for the passengers to return fire; but moving at convoy speeds, with buildings whizzing by, and only a narrow port to shoot from, odds are all you're doing is "spray and pray".
I suspect a gasoline bomb would render such a vehicle unusable in short order, but I am no expert.
When tossed properly, such as the air intakes for the engine or crew, or at the crew hatches forcing them to button up, an incendiary IED can be effective in a short engagement. To disable the vehicle for longer would take a bit more; blinding the driver's vision is an effective move.
Very difficult in an MRAP since they have full windshields rather than periscopes. I'm sure the tires burn marvelously, though.
They almost never ever operate alone, and one of the main tasks of the vehicle commander up on top with the light machine gun is to “dust off” his buddies by shooting anybody who gets near or on the other vehicles. His light MG isn’t going to damage the armor of the other vehicle, but it will keep that brave person with a molotov coctail out of throwing range.
Yup, and loading the engine intake with ether, propane or other light fuel would lock up a running engine or send it into overspeed. Ever seen a runaway diesel engine explode? I have seen the results and it is a mess to clean up.