Is there any method to output VGA with Arduino?
This is quite difficult to do well but there are a couple of examples. The main difficulty is in the speed of the CPU.
Here's one doing direct VGA output: Make: Arduino VGA Demo
If you Google "Arduino VGA" there are a few demo projects.
A couple options are the "TellyMate" shield that lets you output Composite rather than VGA (the single yellow connector you find on your TV.
Another is the "PICASO Universal Base Board" and "PICASO VGA/SVGA Graphics Controller" - this is not a direct shield but probably the best option I have seen.
Both the above items are available from Sparkfun and Little Bird Electronics among other places.
Yes, but getting the VGA timings right is all about carefully timing your instructions.
I'd recommend the AVGA library, it can output VGA and CVBS/Composite PAL and NTSC from a humble ATMega168. It handles everything under interrupt, so you can just write your application in C and not worry about VGA timing.
It even does sound.
I built a little pong game using it: http://blog.hodgepig.org/2009/08/17/510/
I'd suggest using a PockeTerm kit from Briel Computing to interface with the serial port of the Arduino. It's a Parallax Propeller-based system that acts like a VT100 terminal, with VGA output and PS/2 keyboard input.
Plus, when you're done with that project, you can reuse the PockeTerm or even load other Propeller software on it for experimentation.