The answer to this one would be mostly. By that I mean that a modern distrib (i.e. one released in past year or so) will have all the same libraries, compilers etc. Generally you shouldn't have any problem with this one. Especially since slackware (the oldest active distribution and somewhat of a laggard) is finally moving to the latest library versions with everyone else. Occasionally you might have to tweak something, but nothing like having to have separate source trees.
I'd say worry about it if you have problems, it's really not that big a deal.
2)If I make a program for KDE, will it run just fine with Gnome?
If they have the KDE libraries installed, but are running Gnome as their desktop, you should have no problem running KDE stuff. But if they do not have the KDE libraries installed at all, then no KDE stuff won't run.
For example I run gnomeicu under KDE because it's a much better ICQ program than the KDE ones available.
3)Can a program made for Linux run on Be and/or FreeBSD?
FreeBSD - yes.
Be - maybe, I know that gtk & SDL will run on Be. So you can compile a gtk+ app or an SDL app and run it on Be fairly easily (theoretically, I haven't tried this). I know Be has a POSIX layer, so almost all command line stuff will port fairly easily. As to anything else, I'm really not sure.
4) Any particular difference between Linux, BE, and FreeBSD?
Ummm, I don't think we have time for that question right now. It's kinda of a biggy. I guess I can give you the dime tour though.
a)Linux - Stable, responsive, Good->Fair driver availability, reasonably secure. Linux is a good general purpose OS, it can handle almost anything you throw at it. Linux is developed as a distributed benevolent dictatorship.
b)FreeBSD - Very stable, responsive, Fair->Poor driver availability, more secure. FreeBSD is a very good server OS, but lacks a lot of the HW compatability that is more vital for the desktop. A very good OS, but I don't run it because it doesn't support as much HW as Linux does. Plus it nice that more stuff is prepackaged for Linux. FreeBSD is developed (primarily) by a small committee of people.
c)Be - I have not run Be myself, but from what I've seen it looks interesting. Lots of multi-threading, excellent multiprocessor support. Tight & Lean is the theme I guess. OTOH, it has even less drivers available than FreeBSD. If you want to run exotic HW, this is not the OS for you. Be is developed as a SW product for sale by Be.