1. A PIC can be programmed using assembly language, C or BASIC (or any other language for which a compiler exists).
2. No. That header is used for ICSP (In-Circuit Serial Programming), which is very handy, but you can unplug the chip from the breadboard, plug into your programmer for programming, and then plug it back into the breadboard.
2b. Absolutely yes. I've been using the USB version of that programmer for years to program PICs. It's a beauty. Notice the white header on the right of the board. That plug is connected to the programmer header on that breadboard schematic you linked for the aforementioned ICSP programming. The cable is included with the programmer.