The title is the question. If the answer is yes, then is there a way to retrieve the product key from a dead PC? I do have the recovery CD.
Yes, you will need to supply your Windows 7 key during a repair installation (according to this guide).
To retrieve your key, you could use this utility. It offers the functionality of retrieving a key even from unbootable windows partitions. The only downside is that you need to connect your drive to an other, working Windows machine and then run the program.
There might be other piece of software (which might run from USB sticks, or something) that would enable you to search for your key without plugging your drive out, but I haven't used anything like that myself.