You can test the fixer. Cut off a small piece of the film leader and drop it into a small quantity of fixer at the correct dilution and see how long it takes to clear the film. If this happens within say 1-2 mins then I can't think of any reason not to use it unless someone here knows different.
As far as paper is concerned, if you get a very slight grey look from developing and fixing an unexposed piece of paper then you can try some benzotriazole in the developer which may clear the greyness.
A bigger issue with old paper might be its loss of contrast. This can be tested with under the lens filters and trying to get progressively more contrast in the prints. If this fails to happen then the paper is probably for the waste bin.
However if I were starting to print and had to use commercial darkrooms where time is money and good early results are important for learning feedback then I'd invest in fresh paper.
Keep the Barclay paper for now and try it out after you have more experience