That router in that small of space should be fine. Sounds like something is blocking the WIFI. Is there something metal in between? If you move to the doorway is the signal good? Can you borrow...

I would think at least the repeater would have to be a client of the WIFI network so it would be seen.

There are different ways to based on what you are trying to achieve. I played with two routers a few years ago using different brands of routers (a Lynksys "g" and Netgear "n" router). I do not...

Because people get brand loyalty the same as they do for cars.

First, ensure the speed from your ISP by connecting a PC directly to the modem bypassing the router via cable and run speed tests. Assuming all is good... Try moving the router as even a few feet can...

There is also a common thing called a repeater which is another option but since you have the cable in place, an extender is a better option.