The answer is yes, although you must have left the prints in the fixer for quite a while to see the staining immediately. It's usually something that creeps in over time. It may be the case that your paper has oxidized from the edges inward before you used it, but you can check that by simply wasting a sheet and processing it "by the clock" (without exposing it, so you can eliminate as many variables as possible).

The length of time in the fixer isn't super-critical or anything. It's only the developer that you've got to get bang on; the stop bath and the fixer are (as you surmised) more or less a meet-the-minimum sort of thing. But the fixer is reactive, and for prints that are meant to last more than a very short time, it's something you want to get rid of. Otherwise you'll get that loverly yellow/brown patina of age in a hurry without the bother of toning.

In a typical "intro to the darkroom" course, you'll be shown the things that are necessary to make magic happen -- straight develop/stop/fix. And if you were shooting old-tyme news photos in the Weegee style to get something to the prepress guys, that's all you'd need. But if you want prints to last for any time at all, there's still a step to go, and that's to get rid of all of the residual chemistry (mostly the hypo/fixer). Your prints need at least a wash in running water -- how long depends on whether the paper is fiber-based (long time -- typically five minutes or more) or resin (shorter, since the "paper" itself doesn't absorb much and you only need to "clear" the emulsion). A hypo-clearing rinse (compatible with your fixer, of course) before the wash can speed the process up considerably.

An "archival" print washer is fairly cheap and easy to make if you can't pick one up cheaply enough. There are several DIY solutions online -- just search for "archival print washer". Both the "cascade" and the "upflow" desgns are effective; pick whatever looks easiest to make to you.