Both recording checks and draw offers are personal preference, and neither is required in your notation.
I actually do record both checks as "+" or "++", and draw offers as "(d)". I know I am not the only one to note draw offers as I have friends, who do the same. I also record times.
The purpose of the notation is so you, or the arbiter, can play back ...
According to the FIDE Laws of Chess:
Appendix C. Algebraic notation
C.12 The offer of a draw shall be marked as (=)
As with a number of the laws regarding recording of the moves this is not strictly enforced by arbiters. I did once play in a tournament in which a very junior arbiter delivered a short harangue to us players telling us to record draw ...
Yes, a double check is when two pieces give check at the same time. It doesn't matter how you got to that position. As to whether there's an error on that wiki, it's probably best to try to ask its author. Maybe they can provide the full list of positions for you to compare.
I can think of two ways of giving double check while promoting. One is where you ...
I am pretty sure I could make this last longer, but here is already a sequence with 26 unforced consecutive double-checks without promoted pieces.
[FEN "8/8/8/Rp2p3/3Pk1N1/6pB/1Q2N2B/K3R3 w - - 0 1"]
1. Nc3+ Kxd4 2. Ne2+ Ke4 3. Nc3+ Kd4 4. Ne2+ Ke4 5. Nc3+ Kd4 6. Nxb5+ Kd5 7. Nc3+ Kd4 8. Nb5+ Kd5 9. Nc3+ Kd4 10. Ne2+ Ke4 11. Nxg3+ Kf4 12. Ne2+ Ke4 13. Nc3+ ...
Yes, the case you mentioned would also qualify as a double check.
But it also depends on what your definitions are. If the Perft result does consider that to not be a double check, it doesn't mean you have to abide by it.
Yes. I was able to locate a game between grandmasters which included a double-check promotion.
[Event "17. ch-EUR Indiv 2016"]
[Site "Gjakova KOS"]
[White "Ter-Sahakyan, Samvel"]
[Black "Ponomariov, Ruslan"]
In the time since @Evargalo posted their answer, a new record of 14 forced double checks without promoted pieces has appeared in a selfmate problem.
[Title "Miodrag Mladenovic, MatPlus.net Forum 14/11/2019, s#15"]
[FEN "BQN3N1/1RPRPk2/1B6/1b4np/2nr3r/4q3/4P1p1/4K3 w - - 0 1"]
1. e8=Q+ Kxe8 2. Nd6+ Kxd7 3. c8=B+ Kc6 4. Rc7+ Kxd6 5. Rc6+ Kd5 6. Rd6+ Ke5 7. ...