Salting the water in which you cook starches (pasta, rice, potato) is an effective way of enhancing the flavour of the finished product - boiling starches absorb salt well (which is why adding chunks of potato to an overall salty stew will lessen the apparent saltiness of the dish.
But salt does other things. When I am making roasted potatoes, I parboil them for 5 minutes before drying and roasting them in oil. if you divide them into two batches and boil one half in unsalted water and the other half in well salted water (1tbsp/2 quarts water), the salted potatoes will brown and crisp much better than the unsalted ones. I'm not sure why this is, but I would encourage you to try it because it's amazing to see.
Beer, like wine or coffee, is often used when a reduction over a long cooking time is called for but water would be sub-optimal. I make chili a lot. Water is not your friend there, especially if you incorporate a lot of elements that have water in them to begin with (undried/roasted tomatoes in particular). Beer doesn't add the acidity that coffee does, but it is great for adding sugars and the maltier flavors. The sugars aren't precisely 'sweeter' but rather add depth of flavor.
I'd be interested in the effect of carbonation on the cooking process, however since carbonation is supposedly lost faster at higher temperatures I'd imagine it's less than expected.
Beer can be used to de-glaze and so on; its lower alcohol makes it much less reactive (than marsala or liquor) and the sugars make it as likely to glaze over. Still it imparts some flavor depending on the context.
The type of beer you should use is largely up to you. They differ greatly in flavor and composition. For some suggestions on pairings check out this question. Another consideration is to look into Cicerones; it's a certification people get where (more for large scale operations than restaurants or personal use), in addition to knowledge of processing and manufacturing, they are sommeliers of beer.
An additional consideration is how long the beer will be cooking down. The longer the cook down, the more the flavors will be less distinct. However, there are flavors that are frequently only found in beer; hoppiness, roasted malt, and (the effects of open fermentation with) wild yeast can do amazing things in a dish. Creating a side reduction minimizes the cook down and can preserve the flavors.
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I can think of several reasons why you might salt beer:
I've heard of it being done before, but never with good beer, only low quality swill. But then, since there were more quality problems with cheap brews at the time, this was probably a lot more common.