1. If you hadn't eliminated the router from the equation by hooking directly into the DSL bridge (it's not really a modem, but I won't argue semantics too much
) I would say that it could be either the router or the modem. Since you did remove the router from the loop though I would say that it's doubtful it's causing the issue although I'm not entirely certain it's necessarily the DSL modem.
2. It would. When you change the modem to bridge mode it stops acting as a DHCP server and doing any routing duties. I may have misread it but did you change the settings on the modem back to the default when you hooked directly into it?
3. Routers are really nothing more than highly specialized PCs. Your WRT54G actually has a CPU, RAM, NVRAM (for storage) and an operating system. So it's possible for them to have issues with the operating system that require a reset of the router. I don't have it happen very often with my Linksys routers but it can happen.
4. I doubt that it's your Xbox affecting your entire connection since it sounds like it also happens when you're not using the XBox. The easiest way to make sure that you have open NAT show up on your 360 is to establish a DMZ and set your XBox to use the static IP that's assigned to that DMZ. However I very much doubt that it's the cause of your problems.
5. It's pretty tough to narrow it down between the modem and the service provider without replacing the modem. I would recommend giving your service provider a call and see what they can do to help resolve the issue. Make sure you tell them that you've already attempted to bypass the router in your setup and the issue still occurred. Every single ISP that I've had to go through for tech support I have to do this since it's the first thing that they ask, that along with resetting the modem. There are a few possibilities though:
1) Your modem is having some issues and needs to be replaced
2) You're on a higher speed plan than what your line can actually handle. What speed plan are you on and do you know how far you are from the CO?
3) There are issues with the line itself either inside or outside of your household, or some of the equipment that it goes through before reaching the CO (DSLAMs etc.)