1. For troubleshooting and repairing systems, what are some good (free?) applications that I could carry with me on a thumb drive?
AVG, HijackThis, CCleaner, Panda Cloud Antivirus, Windows XP/Vista disc, Linux Live disc is always good for rescuing files from a corrupted / dead-ish hard drive before reinstalling.
2. What tools should I absolutely make sure I have on me when going on a call?
Good set of screwdrivers, hex set, canned air, small brush to rid of dust. Large thumb drive or external HD.
3. What kind of supplies (cables, hardware, etc.) should I make sure to have on hand when going on a call?
Network cables (maybe keep a bulk roll on you and cut it as you need it), people always need wireless routers so it wouldn't hurt to have a new one in the car, new (sealed) hard drives would be good, mice/keyboard sets, etc.. this all depends on how much money you're willing to invest upfront. Always go to NewEgg for the best prices.
Even if you are missing something just be polite and explain the situation. Do not rush, even if you are rushed. Communication is key to keeping customers and making them happy. And always hand out a few extra business cards.