Yes, you need new pad. Think of carpet pad as the foundation for your carpet....the stronger the foundation, the more durable what's above it. The better the pad, the longer your carpet will last. You can put an inexpensive carpet on a good pad, and ...
Go for carpet. Get a carpet rem, it will be cheap. You can get something you like later if you finish the area. Do not paint, It will be hard to put down tile or lino if the concrete is painted. You will need to remove 99% of it before you do.
A second vote on the goo gone. I used the orange spray kind of goo when my kids got gum on my carpet. It worked great....you would never know there was gum stuck in my carpet. Also, I used a warm/damp wash cloth with a bit of carpet cleaner when I wa...