Yes, products can be in multiple categories. They function like normal categories do. For V3 (out tomorrow or Wednesday) because the split blog / shop now come by default, you'll need to create a parent category called “products”, and you can have as many subcategories as you want. For example..

Products

- Nike

——> Tee Shirts

——> Shoes

——> Etc

You can have category images, you just need to change the code a bit. There are plenty of tutorials all over the place on how to easily do this.

Can you point me to some tutorials regarding category images or are they here in the forum?