Kumarb,
The tray corrosion is almost certainly caused by spilled battery acid. This comes about in any of several ways. First, if you are refilling a battery, when you add the water to the cells and the cells overflow, what overflows is acid. Second, those caps on the cells are vented, so hard stops or fast starts can actually spill small amounts of the acid out. Older batteries, or those that have had the hold-down too tight can also crack--thus leaking the acid. Lastly, certain charging conditions can cause the acid to boil or bubble in the battery, thus also spilling out. Any time you get what appears to be water coming from your battery, it is actually acid and it goes in and around the battery tray, causing the corrosion you note.
What to do? Well, first--if you are not going to show your car in an MBCA Concours where originality counts, get a sealed Optima Gel battery and your problems will be over--expensive but worth it. If you continue with a lead-acid battery, you can purchase a "mat" that fits under the battery. You can buy these at any auto parts store for a few dollars. The mat is like a thin sponge and is impregnated with "basic" chemicals which neutralize the acid and minimize corrosion to some extent; you should change them annually. You can be careful NOT to overfill the battery cells. You can, annually, remove the battery, and neutralize/clean the battery tray. Baking soda and water or dilute ammonia will do a fine job of neutralizing followed by a thorough rinse with clean water. If you are concerned about water, be sure to dry the area thoroughly using towels and either compressed air or even just one of those small electric leaf blowers before you reinstall the battery. Be sure not to spill any acid in the reinstall process!
Your new battery tray is probably painted. This isn't as durable a surface as powder coat; you might consider powder coating the tray and the hold down as well. This offers a greater measure of protection against the accidental spills. I've done my hold down and my battery tray is next.
Hope this helps and good luck!