Hi, Yes your Tahoe came with Keyless entry from the Factory. As the other poster suggested, the Alarm most likely crapped out, and was somehow bypassed-shut off.
Tough to say what type of alarm, but there will most likely be remnants of it under the hood (Siren possibly), and under the dash, and will no doubt make use of a stock Keyless Entry controller impossible, until this system is completely removed from the truck. They no doubt spliced into wiring that controls the Door Locks, etc. Look for little black boxes near the base of the Steering Columm-under the Dash, with names like Clifford-Avital-Viper-etc will be a sure clue.
Best-cheapest option is probably going to a good Alarm dealer-installer, have them remove whatever's left of the old alarm, and install a new entry level priced Alarm, with its own Keyless Entry Fob.
The Chevy Dealer would no doubt take you to the cleaners $$$$$$.
As for the Windshield Wipers: Before spending any money, use the proper Torx-Head Bit with Screwdriver, and remove the Black Wiper Control Box Cover on the Driver's side of the Firewall. Clean down the entire Board with Contact Cleaner, if grease is laying all over inside, and on a couple of round Metal Discs, re-distribute that grease onto the Worm Gear visible. Do the best visually inspecting the Board. The usual culprit was the Plug, and solder connections that went to the Board Just leave a very light thin coat on the Metal Discs. Re-assemble, and give it a try.
I myself was beginning to get the same erratic problems on my '97 Tahoe a couple of years ago, and luckily this simple fix-maintainence cured the problem for me.
If that doesn't solve the problem, you'll need the new Board from Chevy and do a swap out. Yes, at one time I believe there was a safety recall, but I believe the free replacement through GM is long gone. Hope this helps. Mark