Yesterday I rearranged the room my router lives in and had it unplugged for about half an hour... since then it's lost its mind. Here's my problem(s).

I can no longer set Security Methods (before I had a WEP). It reads "Disabled" and when I open the drop down menu to select security it won't like me actually pick one. If I go to the Wireless page (this is all under router settings at the router's address) and reset security, it will turn on WPA Personal. It never lets me pick another setting. It also changes the name of my router on SSID to something random. If I change the name back it seems like it causes problems with either the WPA or the wireless computer seeing the network.

Also, the wireless computer can't see the network show up. It does once in a while, but it constantly disappears. It seems related to when I try to mess around with changing the name back, or reinstating security. I'm not sure. This thing was totally plug and play before (new device / computer always saw the network, just bang in the WEP and good to go). Finally, when I try to reset my password to access the router page, it never acutally asks me to input it. Is the router on its way out, did something malicious get in or am I just missing something?

It's an older Linksys 54G - the thing hasn't been a problem before now. And no there are no firmware updates :(

Any questions ask away as I know I'm not being too clear. Help me out here! Oh and yes I've unplugged it again and tried resetting it with the button. Many, many times... grrrrrr!

One suggestion, though perhaps a bit more advanced than you'd like, is to try flashing it with dd-wrt. Will give you a lot more control over the router than the original firmware gives.

Along the same lines, I've been using Tomato for ~6 years on my WRT54GL with absolutely no problems. Check out the Toastman builds, they add a ton of features to your router (probably very similar to those in dd-wrt).

Starkrun's link is an instruction for how to reset the router to factory settings, the same instructions are at Linksys' site, here: http://kb.linksys.com/Linksys/ukp.aspx?pid=80&app=vw&vw=1&login=1&json=1&docid=26b93b8835dd4fb595962dab15771be9_19723.xml&

DD-WRT and Tomato (as itchanddino mentioned) are commonly used replacements of the official firmware for a lot of routers. Wikipedia links, in case you want to learn more:
https://en.wikipedia.org/wiki/DD-WRT
https://en.wikipedia.org/wiki/Tomato_%28firmware%29

If you're wary of replacing the firmware, resetting the router is the first thing I'd try.

I did a reset a few times - same problems. If I do try and flash some new firmware, will it eliminate anything malicious that may or may not have gotten into the existing firmware? Are these things legit to download from sites (tomato, DD).

Also, if I do just break down and buy a new router today / tomorrow any recommendations? I don't need high end (I played a little LOTRO a few years back and had no problems with my performance and the wireless computer is just used for browsing) but are there any out there that are a major pain to set up / perform? Are there any that stand above the rest?

All right, reset done. Good work.

If anything malicious has been inserted, you might even try reflashing the most recent official firmware. If the router has its own automatic update function (ie. click a button in the web interface, and it'll do everything for you), that won't work (since it'll simply detect that it already has the latest) but you'll have to do it yourself. Instructions: http://kb.linksys.com/Linksys/ukp.aspx?pid=80&vw=1&articleid=4030.

And yes, both DD-WRT and Tomato are safe to download from their respective web sites - and some manufacturers even support them and others officially on some of their models (Netgear on their WNR3500L, for example, which I bought for that reason, because I like to tinker, and Buffalo has shipped it by default on some models) - but since both support hundreds of routers with different hardware you need to know your exact model so that you don't try to flash a file that won't work on your device. For example, there's not only the WRT54G, WRT54G2, WRT54GS and a few others with another last letter or digit, there are also versions of each, most are supported, some are not yet, and still others won't be (for example, DD-WRT doesn't support the WRT54G v7.0, but does support the v6.0 and earlier as well as v7.2 and later)... I did say it might be a bit more advanced than you'd like.

Ah awesome I might give this a go as a last ditch effort then, knowing that these firmwares are so legit as to be officially supported.

And luckily my 54G is V5 so it seems like I'll be good to go if I try it!

As for what router to choose, should you get a new one, I don't actually know. Linksys and Netgear are well-known brands, I heard some Asus routers were good, and I don't know if Cisco still sells consumer devices since they sold off Linksys to Belkin. I also don't have any experience with Belkin's devices.

Better would likely be if you found a few routers within your price range in stores you trust (whether that's ordering or in your area), then we can likely help you find info and reviews/opinions on them.