Rim lock yes
My DRZ as well as every dirt or DS bike, came with a rim lock. You should have one on there to keep the tire from spinning on the bead at normal tire pressures front and rear. The rotted rubber was probably the rubber band that wraps over the spoke nipples. You can replace it, or just run a couple wraps of duct tape around the rim over the spoke nipples.
If you are running on the street, its worth balancing the rim locked wheel with weight opposite the rim lock. Balance with the wheel on the axle between 2 jack stands. It will make for a much smoother ride on the road. I just use stick on wheel weights with a bit of silicon to help keep them attached. Ugly, but functional, and easily replaced when they get knocked off by trail obstacles. There are prettier and more precise options out there, just ask the SM guys.
First tire change is a big step! Doing the first one solo, without an experienced guy helping and without trip to the ER is quite an accomplishment, and you should be proud.
The Clymer manual for the DRZ is pretty darn good. Just make sure you are looking at the section for your correct model when it comes to carburetor, electrical, suspension, etc.
02 DRZ400E Dualsported in AZ
14 KTM 250 XCW (Plated in AZ)
85 HON XL80S (Wife's ride)
00 and '07 YAM TTR 90s (kids' bikes)
89 Lockheed F-16C (Work vehicle