1) Is it required to have a persistent custom HTTP header in all requests, e.g. "X-Bug-Bounty: {bitcointalk username}"?
No.
2) Is it acceptable to use newly created / generic sock puppet accounts for testing?
Yes.
3) Is the "The Glider" forum badge assigned in all cases where a vulnerability is disclosed and patched, or only when a payment bug bounty is provided? (I am curious if this badge will be given out for low to medium risk findings that are not eligible for a payment bounty, but could still be useful)
Probably only for the listed security bounties.
4) If my genuine IP or testing accounts are banned for suspicious use whilst performing bug bounty testing, will my normal BitcoinTalk account remain unaffected?
IPs are only banned for making too many requests, not for suspicious behavior. So just don't make more than one request per second.
A lot of people try to run "website security scanners", and then report the "bugs" that these software packages find. Please don't do that. These scanners only ever report little configuration things which some people consider
less than ideal, like allowing certain TLS ciphers, or sending/not-sending certain HTTP headers, and stuff like that. It's not useful.