It's reasonable to exercise caution with asbestos-containing materials, but I think you are somewhat over-reacting to the issue of the asbestos that may have been improperly ripped out. Any of the fibers would have been long gone years ago as they are incredibly light and would have been wafted away. Even with doors and windows closed, homes and apartments have several complete air changes every day and this would have the removed the fibers. As for the risk from the remaining materials, I think you are being prudent to simply seal them off, although I don't see them as a significant source of fibers unless you are disturbing them. While these kinds of things are no longer installed (and, in fact, I think they are banned from manufacture), even with a fire going, the majority of any released fibers would have gone up the chimney.

Bottomline: it's good you are aware, but you should have no concerns about bringing contamination with you to your new place. Exposure you have had from this source over the years was probably minimal, and you need not be concerned that this problem will follow you in some way.

I have read a little on this topic due to the fact that some varieties of soapstone, which is used as a countertop material, can contain asbestos. Some people use soapstone for carvings, and there is a bit of information on another website about someone who suspected he may have been exposed in that manner. I am going to give you the link to that page so you can read more for yourself. There were various suggestions, including contacting a lawyer to go after the people responsible (in that case, for selling asbestos-containing soapstone; in your case, for the hazardous rental environment). The idea was to thoroughly clean everything after exposure, but if the fibers have gotten into clothing and furniture, you could be looking at a serious issue. You might want to try contacting a university in your area or even state government officials for help. One of the first steps would be to get the material tested to ensure that it actually is asbestos. You should have a lot of resources available to you at the state level. Also, the community not only in your building but also in the surrounding neighborhood, needs to be warned by public health officials. Here's that page where you can read more about asbestos in the situation I mentioned above: http://sculpture.net/community/showthread.php?t=8511

Belaria- There’s also asbestos in vinyl asbestos flooring tiles. They aren’t manufactured anymore, but they’re in millions of houses and don’t pose a risk unless you start destroying and sanding and grinding them. If there is asbestos in a soapstone countertop, it might pose a hazard to the fabricators If they don’t wear respiratory protection. But the likelihood of fiber release in your home is trivial unless you decide to sand the countertop. By all means be cautious when dealing with asbestos containing materials, but the idea of contacting a lawyer and seeking some kind of damages in the situation the OP posted about is likely to cost a lot of time, money, and stress, and not go anywhere.