Which model and do you have the latest firmware installed? You might be dealing with a firmware bug.
Same here with my Linksys BEFW11S4v4. For a long time, I thought it was something inherently wrong with the firmware or hardware. Every few days, I would find the router section effectively hung. Lights were ok, ethernet to wireless bridging just fine, but the router was comatose. Cycle the power and it's fixed.
Eventually I found the cause. I was getting attacked from the internet. The attack was hanging the router. I started sniffing my incoming traffic and found the attacks. My router is particularly susceptible because it fails a few router exploit tests.
Same here.
You've performed a "reboot" of sorts. Nothing wrong with doing that.
That's wrong. It's safe to unplug your router for extended periods. Your unspecified model Belkin router has NVRAM (non-volatile memory) that will save your settings literally forever. There is no battery inside that might discharge itself if there's no AC power applied. In other words, you could leave it unplugged indefinately, and still retain your settings. Most routers also have a "save settings to disk" feature that allows you to backup your settings. If you need to reload for some reason, then just upload the saved settings. I use this feature for paranoia, but rarely use it.
Sigh. That's a difficult choice because the answer is half way in between. The router should be able to run forever, without crashing or getting clobbered from the internet. At worst, it should have a "heartbeat" circuit that detects that "something has gone wrong" and automatically reboots the router. All my SCADA hardware has this feature including the PC's. You should be able to plug, play, and run (away).
However, life is not so simple for a cheap router. The key word is cheap. My Cisco, Sonicwall, and Netscreen routers will stay up forever. I don't power cycle them after firmware updates. It's considered a disgrace to call the NOC and have them power cycle a router. However, these are expensive routers from companies that consider 24x7 operation a key specification. They will spend the time and money to insure that uptime is maintained. That's not the case with Belkin and other cheap routers. If there's a firmware bug or vulnerability that requires an occassional reboot, most home users can live with it. It might get fixed in the next release, maybe. However, with product cycles measured in months, there's no incentive to make a long life product. So firmware updates stop after a few months and the initial bugs become permanent.
This is one reason I detest Belkin. Their firmware updates generally stop at the initial release and never extend for more than about a year. Linksys, Dlink, and Netgear all consider it important to update their firmware for the latest features and to fix outstanding bugs, issues, vulnerabilities, and stability issues on obsolete products.
Anyway, feel free to turn it on and off as many times or at any interval you feel necessary. I don't think one power cycle every few weeks is doing all that bad for a cheap router.