You might be able to place the amp behind one of the seats... but you need to use a mounting board (1/2" or 3/4" MDF) between the car and the carpet. You will lose some leg room... but if that's not a problem for you then it's a possibility.
You won't be mounting a sub behind the seats unless you are 4' nothing and have the seat all the way forward.
You can use pretty much any sub with that amp. The big thing will be: can the sub HANDLE the power you deliver to it? It's always okay to have more power on tap than the sub can handle, so long as you are SMART about the gain settings and the volume knob.
Using sound deadening in our cars is one of the BEST things you can do to improve the overall audio experience. I won't say that Dynamat is better or worse than any other brand (Except EDead: Stay away!) but while you are looking at dampening material, look into sound absorption material as well. Closed cell foam does wonders for keeping out the noise... look for a product called "Ensolite" from RAAMaudio.
http://www.raamaudio.com/cgi-bin/index.cgi
(Rick is the owner of Rammatt and is also a Forum member!

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I used the spray-on adhesive that he sells (V&S) and it's the best stuff I have ever used. Very sticky stuff and easy to use. I haven't used his RAAMmat but I hear good things about it.
As for the cost... I couldn't tell you. I do all of my own labor and fabrication so I couldn't tell you what it would cost. A typical C5 box can cost about $150, but I build mine. A C6 box might be a bit more than that... take a look for a good deal here on the forum.
Figure that you will need an amp kit, the mounting board, pull the passenger seat and run the wiring, remove the radio bezel and hook into the stock head unit for a sub signal... that's a little bit of work. I wouldn't trust it with a shop that didn't know Corvettes, as many can attest to here. Odds are, if you are handy you can do it yourself with our help. You might even have someone close to you from here that can help out. It might cost you a frosty beverage or two...
Good luck.
Mark