1. Sony DVP S330. Maybe about 5 years old. Plays fine, but it won't
eject a tape UNLESS I HOLD the door open. Otherwise, it works
perfectly. Is there an easy way to fix this door myself?

This is a DVD player, not a VCR. The problem may be related to a limit switch
that is supposed to signal the player SYSCON that the drawer is fully open when
actuated. You will have to check the limit switches on the drive assembly to
find if there are any switches that aren't being actuated or aren't working.
After repair, prevention of this problem is closing the drawer by pushing the
OPEN/CLOSE button, and not on the drawer itself.

2. Samsung VR5901(maybe about 11 years old!). Worked fine until
recently. It recently ate a tape. After opening it up, I can see
that the tape is being "held" in the play position, even with I
haven't pushed PLAY. It will not let me play the tape, but if I push
FF, I can "see" the video going forward; same with reverse.

Sounds like a mode switch problem. The Samsung VCRs of the 90s used what is
known as a fast-load system. The tape is kept fully threaded at all times
whenever it is in the VCR until you EJECT it. This is supposed to reduce the
time it takes for the machine to go into any of the transport modes. However,
a faulty mode switch can cause the VCR to do some funny things, like eject
without spooling the tape back into the cassette first. Another possibility
could be the power supply, believe it or not. Samsung VCRs of 1990s vintage
with switch-mode power supplies can develop bad electrolytic capacitors in the
secondary side over time, which can also cause odd behavior.

Hope this helps. - Reinhart