O.k it breaks down to this
There's the pickit2 programmer and then there's are different demo boards packages you can get with it (starter kit, debug express, serial analyser, ... )
The programmer is the same for each one. This is all you need to do the programming and debugging. The demo boards just help you get a quick start making a circuit and fits in with the lessons. You can program and debug all the different PICs as listed in the link in my first post (pretty much any PIC). The pickit2 programs the PIC by ICSP (in circuit serial programming). With ISCP you can design whatever circuit you want and than program the PIC in the circuit, all you have to do is include a 6 pin header for the pickit2 to fit into (and connect the header to the PIC properly, which is easy). The demo boards do this for you, but once you are design your own circuits you won't need them.
So in short, yes you can debug the 16F877A and you don't have to use the demo board.
Hope this helps