Is there some particular reason not to put new holes in the bricks? My first inclination is to just drill new holes, and secure the spouts properly - after all, they did come off during a snow storm.
Do you have pictures of the existing mounts? Perhaps there is a way to retrofit something onto the existing hooks but still use a modern downspout.
I would venture a guess that @Tester101's suspicion about your floaties coming from your glass (unrinsed soap, dust, etc.) may be heading down the right path. I would try rinsing a glass, scrubbing it with a new sponge or clean washcloth, wiping it dry with a clean towel (not paper towel, as this may leave pieces of paper behind), and then refilling it. @NiallC's comment about asking for a water report from your municipal water supply is a pretty good alternative, but doesn't factor in anything that may come from the pipes between them and you. Most health departments will do water testing, so you could always fill a bottle and take it in.
Beyond that, if you really have to know what's in the water, your only real solution to absolutely know with 100% certainty (technically, unless you watch the health department do the tests, and can verify their results, you can't KNOW that they're right...but I'd probably just accept it) is to get testing kits for this, that, and the other thing (that was just the first page offering a wide range of kits that I found), that operate similar to what you might use for pool water. Depending on the sensitivity levels, you may pick up on things like trace amounts of chlorine...but of course this is going to get back into the more expensive end of things, as those kits aren't cheap (Grainger's seem to run around $40-$50 and up...it's about $20 for their cheapest, which is testing for copper, on that first page of results).
You could pick up a pool test kit, which will cover a couple of parameters (chlorine, calcium, hardness, etc.), but it's not going to cover everything and may not operate in the range you need to identify your floaties...it's designed primarily to help you maintain clear-looking pool water that you're not drinking, not crystal clear tap water.
So really:
Ask for report from municipal water supply
Take in sample to health department
If you're still paranoid and want to know what it is, buy test kits
If you just want to get rid of the floaties, filter, filter, filter, and make sure your dishes aren't the source
Best Answer
Step one - look. Lead is unlikely inside the house, but you might see a lead pipe coming in from the street, if there is one. Copper pipes soldered with tin/lead solder are pretty common. But so are threaded galvanized iron pipes.
Step two - ask the water supplier, if it's on a municipal system.
Step three - have the water tested - it won't tell you if the pipes or solder have lead, but it will tell you if you need to worry about it. Most municipal systems that have lead distribution pipes in service take care to adjust the water chemistry so the lead stays in the pipes, not in the water.
If it's on a well, the odds of lead pipes go way down, IME. But the odds of water chemistry that might release lead go up quite a bit, depending on the water source.