If you have Wifi, use either the Acekard 2.1 or Cyclo DS iEvolution with the DS Lite, and try this tool: http://projectpokemon.org/forums/showthread.php?12405-A-tool-to-backup-HG-SS-saves
If you don't have Wifi, you have insufficient hardware. (Let's get into that after you respond with whether or not you have Wifi).
No. Change the Primary. If the system cannot connect to the Primary DNS, then it uses the Secondary DNS. So, if the Primary is valid, it will never connect to the secondary.
An Action Replay can't backup the save files of HG/SS/B/W alone. If you are using a DS lite, you could use that and wifi, but other than that, you could try an NDS Backup Adapter Plus (or another backup unit).
I am not aware of a way of editing the save file entirely if you don't have a backup tool, but if you want to just edit the Pokemon you can try either Shiny GTS or IR-GTS-BW. They should do the trick.
More options for starters, some ability to choose your character, monster houses being accomplishable (sometimes I can't get throug even with cheat codes). Other than that, I don't know how they can improve, apart from small things.
1. Yes, it is random (well, pseudo-random). Probably easier to edit it than to try to abuse the RNG.
2. 65535
3. Yes, just be sure that you change the TID on all other Pokemon as well to ensure you are still its original trainer.