What rights do I have to profit from a software application I write?
That is, if I write something in, say, C++, can I sell it, even though
the language itself was written by someone else? Is the answer to this
question different if I write something in Java (is Java open
source?)? I know the answer(s) could be very detailed, and probably
worth more than the price I'm offering, but please answer with just as
much detail as you think is worth my price. Thanks!

Generally you have full rights to sell the software assuming you wrote
it from scratch and aren't using someone else's libraries (except
those supplied with your compiler).

However, you should check the license agreement of your compiler and
editor for any special requirements or restrictions.

I'm assuming you wrote this at home on your own computer in your free
time.  If you wrote this on a work computer or during work hours while
at work, you are most likely not entitled to sell this software as
your own.