I use some Wyeast yeast propogation nutrient blend, just a small amount scaled from a starter wort back to the volume being used to make the agar medium. Also, malt extract is used, at the same ratio for a starter sg, 1.040 which is 100 gms DME per litre of water.
With regards to the term broth, have never seen it come up in the reading on yeast culturing. Obviously you find it a common term in relation to your microbiological knowledge.
One method with yeast storage is to maintain a small amount of yeast in sterile (not demineralised) water in the coldest part of the fridge. Have a read of Graham Sander's article on craftbrewer.org which outlines this method.
Many people do report being able to revive very old yeast samples, either from slants, commercial packs from Whitelabs and Wyeast and from yeast sediments in bottles. My concern about stretching this is that we are selecting the yeast on longetivity rather than its ability to brew a good beer. Each method does work, but once the yeast starts propogating, do we only have mutated cells. To test for this is going to take more than looking through a microscope. The general concensus is whatever method you use, revive the yeast every 6-12 months, reculture (or reslant) and then store back away.
Commercial yeast strain banks are kept frozen.
There is no known taste test that will definitively say, infected or not. What i do is to sterilise the starter wort for the first two steps (20ml and 200ml in erlenmeyer flasks in a pressure cooker) and do the transfers as aseptically as possible, minimising infections. The only way you can definitively say, no it is not infected is on the grown out agar plate, or by looking through a microscope at a sample of your wort. There are test kits for commercial breweries, but they will not confirm wild yeasts. Often a starter will taste funky, but that is due to fresh yeast byproducts.
Lucky you having access to a microbiological lab. Make sure your friend shows you how to prepare plates, streak them up and then examine them. Hopefully you end up with some good results and can tell us which method will work the best for long term storage in a homebrew environment.