Don't freak out about it. The asbestos scare is blown WAY out of proportion. It takes prolonged and significant exposure to cause any problems.
That said, if you're concerned then get it tested.
But to me, those tiles look pretty new. I wouldn't worry about it.

mogley-
It's good to be cautious around materials that contain asbestos, but you are having an over-reaction based on that scary word, "asbestos." What you did and what your kids did is not going to cause any health issues...even if those tiles contain asbestos. Some old tiles did contain asbestos and some didn't and you can't tell from inspecting them. I suggest you take a small sample to be lab tested. If there is no asbestos in the tiles, you can quit worrying. If there is asbestos, just don't damage them and have them properly removed and disposed of if you ever remodel, but also quit worrying. Even if they contain asbestos, they are likely to be only a couple percent asbestos and the fibres are bound in to the other materials, so they don't easily get airborne. Those tiles also have a coating on them. Unless you decide to sand them or rip them down and crumble them, there will be no measurable release of fibers. Asbestos fibers are often a component of old vinyl tiles, yet people walk on them all the time without ill effects. Be cautious, but quit thinking you are living in a hazardous waste site.

Bottomline: Get the tiles tested if you are still worried; tell your kids to quit abusing the house; and quit worrying. I'm not trying to make light of your concerns; I'm just trying to give you a reality check.