Answer:
What get's more wet while it dries?
Other terms for this handy device include doohicky, doo-dad, and whatchyamacallit.
What can you keep after giving it to someone?
It goes up and down the stairs without moving.
Built of metal or wood to divide. It will make us good neighbors, if you stay on your side. What is it?
I am the fountain from which no one can drink. For many I am considered a necessary link. Like gold to all I am sought for, but my continued death brings wealth for all to want more.