Amigakit.com is very reputable. I've ordered from them three times now. Awaiting my USB Deneb right now.
The A1200 still has the Amiga RGB 23-pin output. You'll either need a monitor that does 15KHz, an old multisync, or a scandoubler/flicker fixer that is 24-bit. Supposedly Individual Computers will be releasing an internal one for the A1200. Otherwise, rely on the RF(ch 3/4) or Composite outputs to a TV.
For the power supply, I recommend getting a cheap ATX power supply and building an adapter or if you aren't technically inclined an adapters from ATX to A1200 from AmigaKit.
You need OS 3.1 ROMs and disks to install 3.9. (You'll also need a CDROM to install 3.5 or 3.9 from CD.)
The Mouse is an Amiga mouse with the Atari Joystick connector on it.
I'd recommmend looking for an accelerator card with memory for it. At least an 030 processor. I think AmigaKit just got some in for a reasonable price but they may be sold out by now. So you'll have a faster processor and more ram.
-Nyle