I'm wondering: can I save a copy of the hidden .bitmonero folder that contains the blockchain and keep it on my computer, and if my blockchain gets corrupt can I just erase the corrupt one and copy over the new one into the same place to prevent having to wait to synch the whole chain again?
Yes, you can certainly backup the blockchain and restore from the copy if it becomes corrupted. You'd then have to sync only what's been added to it since the last copy.
However, you can't do this without precautions, since monerod will be writing to the database regularly. You need to either exit monerod first, or copy the blockchain using mdb_copy.
You can also use a single backup for multiple machines, though the blockchain will not work on a machine with a different endianness (though it is otherwise compatible with different architecture differences such as word size).