What do I do with the "tea" from the bottom of the composter?

Dilute with water and fertilize your plants with it. Leachate isn't
as high quality a fertilizer as brewed compost tea (maybe "compost
beer" would be a better term for the brewed stuff, but tea is what
they call it) or compost itself, but as long as you don't
overconcentrate the leachate, it won't hurt the plants.

Some compost bins (for example the worm composter at
http://www.happydranch.com/canoworms.html ) have spigots at the bottom
to drain off the leachate.

If it stinks, I'm sure my neighbors won't want me to let it drain into
our common slab.

Most composting involves some odor, although it should be possible to
keep it modest enough that it isn't a problem. Putting your leachate
somewhere other than the slab probably would be a good plan.