What you have stumbled upon is demi-glace, and is one of the most useful ingredients in Escoffier's canon of sauces. It is, as you have said, a concentrated powerhouse of goodness, and will give an unparalleled richness to anything you add a spoonful to. Soups, stews, and risotto come up frequently in discussion of its uses.

It's fine.. you could either add some water or just throw it in the fridge as is. Either way it'll do the trick. :)

I am so excited to be able to give this answer, because I recently discovered something great! You can actually turn that syrup into tiny, easily stored (and if need be, transported) bouillon cubes:

http://nourishedkitchen.com/homemade-bouillon-portable-soup/

I just did this for the first time about a month ago after storing my bone broth in glass jars in the freezer. That was ok, but then I'd have to remember to defrost some when I wanted to drink it, and it also took up a lot of space. By reducing the broth into a sticky syrup and adding a bit of extra gelatin at the end, you can pour it into a glass pan, let it cool and then when it is completely solid, just cut it into cubes and store them in a zip lock in the refrigerator. I was worried it would be hard to get them out of the pan, but them spring right out.

They are super concentrated, so when I want a cup of broth, I just pour boiling water over a cube in a mug and wait 2 minutes and I have instant broth. I found this recipe (see link) when I was getting all anal-paleo about having to travel across the country for Xmas, and not wanting to be without my bone broth. According to the recipe, the cubes don't even have to be refrigerated! I just threw a ziplock with some in my suitcase and was good for the whole trip.

It only takes an extra hour or so to reduce the broth after you strain everything out, so it really isn't much more work (and no washing all those jars!) I hope someone else tries this because I was thrilled with the discovery.