Using a usb charger for iPhone 13 does this damage the battery

No. Apple encourages use of the older USB power adapters. See the following from Buy iPhone 13 and iPhone 13 mini - Apple

As part of our efforts to reach our environmental goals, iPhone 13 and iPhone 13 mini do not include a power adapter or EarPods. Included in the box is a USB‑C to Lightning Cable that supports fast charging and is compatible with USB‑C power adapters and computer ports.

We encourage you to re‑use your current power adapters and headphones that are compatible with these iPhone models. But if you need any new Apple power adapters or headphones, they are available for purchase.

No. However, there's different quality out there, but as long as you're convinced that you have a quality power source, I say go for it