A) Assuming the problem is not with the machine, my first guess would be that the clear leader on the tape has simply broken off of the take hub and is now inside the cassette shell. This is the most common problem I see with VHS tapes. The leader is anchored to the take up reel by a pressure fitted anchor pin. A sharp crease is formed on the leader at the point of contact and eventually it just weakens and gives way, normally when the tape is rewound. This is a simple repair and can usually be done while you wait.
A) Yes. Regardless of how bad it might look, I’ve never met a broken tape that couldn’t be restored to working order long enough to at least be copied over to a DVD. Cracked and broken shells are a simple repair. Your tape will be transplanted into a new shell and the repair can usually be done while you wait.