Yes, you can process color film in B&W chemicals. That's one of several types of cross-processing -- other common ones being color print (C-41) in reversal (E-6) chemicals or slide (E-6) film in print (C-41) chemicals.
When you process either color print or color slide film in B&W chemicals, you'll get a B&W negative (unless of course you do B&W reversal processing). I've never done it myself, but my understanding is that most color print films produce a B&W negative with a dense orange mask, but slide film produces a less pronounced mask.
I don't recall hearing about problems with fixer when doing cross processing, but I can't say with certainty that there won't be problems. If you want to try it and you're concerned about this, just wait until your fixer is nearly exhausted and do your cross processing as the last roll in that batch; or set aside some fixer to use only for your cross-processing experiments.