Yes, I have done several in all kinds of vehicles. I have several pressure washer's at work and over the years have done so many I have lost count. All I suggest is that if you use a hot pressure washer, when you let it dry, try to place it so it retains it shape as best you can. ( I have had some loose it shape from the hot water heating the fabric and glue too much. )
As far as dye, I have never dyed a carpet, but I am sure you will get some info from someone who has. Good luck. Tom