Asbestos removal is clear in some cases and not in others.
There are terms such as 'friable', 'non-friable', 'airborn', etc., etc., that come into play when determining if removing the asbestos needs to be done in a space suit by professionals or with household clothing and an asbestos dust mask. It's easy for someone to come into your house and either dazzle you with brilliance or baffle you with BS. You're doing the right thing by questioning all this. What I suggest you do is this: First, contact your local town or township office to see what they say about removing asbestos vinyl and glue. After that, if they don't know, they may refer you to the county office. If they don't know, or aren't clear, contact your state office. And hopefully, you'll have your answer on its removal and won't have to contact the Federal EPA!!!
In my jurisdiction, if the vinyl is glued onto something OTHER than the subfloor, like an underlayment, I'm allowed to remove it myself. Of course, full-body clothing and an asbestos mask and gloves are to be used. And you should curtain-off the area you're working in with plastic so the fibers won't go through the rest of the house. Ducts should be covered, heat/A-C should be turned off, and you want to have a spray bottle of water around to keep the dust down from what breaks. DON'T use a saw on this stuff. The best scenario is to see if you can remove the sheets of underlayment/vinyl in tack and get them outside. There, you can drag it to a far corner of the yard and break it by hand (if you're strong enough). If you have to break it up inside the house first, well, you'll have non-friable (airborne) asbestos. This is why you'll need to cordon-off the kitchen from the rest of the house. What I'm also saying is that I don't need to hire folks in space suits to remove the 'stuff'.
Again, check your jurisdiction and use proper clothing, gloves, and mask if you're allowed to do it yourself. Disposing it is a different matter. Check with your contractor AND shop around by calling private trash haulers. This whole thing may not be as bad as you think. You MAY have the option of leaving the 'stuff' in place but I'll leave that to someone else to address. So to answer the latter part of your question, the answer is yes (re: leaving the linoleum in place), and maybe (re: getting around professional removal).