I get a lot of fabric given to me, as well. I have found that using poly-cotton blends in quilts is fine, although sometimes it isn't quite as easy to work with. You do have to keep your iron set a little lower, though. And prewashing is always a good idea.
I usually do a burn test if I'm not sure what a fabric is.
Also edited to add: for something you really want to be an heirloom, you might stick to cotton. Sometimes the poly blends "pill" more with washing. But my MIL says she has a set of supposedly good, high count cotton sheets that pilled a lot, too.