i use an Old Skool AV to HDMI converter, it's $18 and it works like a charm. As for capture card, i use a Mirabox Capture Card i got from Amazon. It's $40

1. I'm sure there's no problem with that since you'll be the one playing.

2. it is allowed, i don't think there's any big changes while using an HDMI adapter (i use one for mine as well)

3. It is not a glitch. You are only respawning back in the room if you fall there (it's basically a death but you respawn in the same room and not at the entrance of the level)