Yep, you can either use (or not) the routing functions of the AE.
That Netgear DM 111P v2 looks like it's just a modem, so the AE is current performing routing duties.
When you get your new modem/router you get to make a couple of choices.
1) You can only have one router in your network, so either the new modem/router does this or the AE does this. Configure as needed.
2) You can have two wireless access points if you want, turn on/off the wireless either device.
Maybe it would be worth getting some extra cable put into the bedroom (or somewhere at the other end of the house) and moving the AE away from the new device to increase your coverage. While it's possible to just use it as a wireless repeater with no cable I wouldn't recommend this since you also want to use it for TimeMachine duties where speed is more important. While the AE is near enough to silent, you just need to be aware of where you're putting it because of the lights it has and the noise the HDD would make ... personally I couldn't stand them in my bedroom.