Yikes. Okay, here goes (I hope you have a digital multimeter):
1) Wipers: sounds to me like you have an exposed wire or something in your steering column. The wiper switch has all it's wires there, as are the ignition wires. If it only does it when the car starts, my guess is that the wiper wire for ON is touching the ignition wire for START. It wipes 3 times because that's about how long you hold the key in the start position? To troubleshoot this, you have a couple options. A) remove the steering column trim pieces, track down the wire for the wipers, and see if comes even remotely close to the ignition wires. B) it's possible you could have a bad wiper switch C) you could pop the plug wire off the coil (or disable the plugs some other way if you don't have a coil) and turn the car over - see if the wipers go while you are holding the key in the START position. Try holding it in the start position for varying lengths of time, confirm that the wipes stay on while in START, and not just go 3 times. Once you get more info here, post it back and we'll go from there.
2) Dome light: there are actually 3 lights in the dome light, two map lights and then a center light that comes on when the doors are open. Are you saying the two map lights work when the the switches are pressed, but the center dome light doesn't come on when the doors are open? If so, most likely you have a burnt out bulb. Replace the bulb, then try again.
3) Key Fob: so it sounds like the fob doesn't unlock the doors, but it does pop the trunk. If the door switch panels work, then the issue is probably in the controller that the fob talks to to unlock/lock the doors. Do you have a known good fob? And if so, does that one also fail to unlock the doors? In other words, are you sure the fob works properly? If yes, then you are going to have to go digging in your doors to get to the controller for the door unlocking. I think that's where it is located - I'll have to look it up in my manuals.
What year is your car?