Yeah, it can happen anytime, it's worse with the older, non-He washers as they use a serious agitator that will Shred a mouse pad. And you can't stick it in the dryer either, even on low heat, as it'll warp and never lay flat again.
As to the pads themselves, there's multiple different kinds. I gave one to my daughter, it's a hard surface pad, has a hard acrylic shell over aluminium, so wipes easily with anything from Windex to 409 or soap and water, there's also plasticized versions like what mom's use for toddlers at the dinner table, several varieties of cloth and hard foam rubber.
They all pretty much work the same at your budget and needs, you'd be looking at over $100 for the ones the 'Pros' use for competition gaming.
With optical/laser mouses today, unless you are on glass or other optically transparent and smooth surface, a mouse pad isnt as much a requirement as it is a luxury for most users.