Instead of sending my friend 3 Win 7 DVDs, is it enough just to send 1 DVD and then email 3 product keys? Also, would a Win 7 Professional product key work if the DVD was for Home Premium?
You can use multiple product keys with just one DVD (as long as they are the same version, see second part). In Windows 7 I believe it prompts for a key after installation, meaning that it doesn't matter what DVD you used... Even if it prompted during installation the product keys are not DVD specific. You don't even have to enter a key until like thirty days after installation. So you can send one DVD and email 3 product keys: the only downside is that you have to wait for an installation to complete to use the DVD for another installation.
To answer your second question: No. You cannot use a product key with a disc of a different version. The Win7 DVDs aren't like the all-in-one's for Vista (in Vista you could install any version from any DVD). Now the win7 discs are tailored to install a specific version, in your case, Home Premium.
The discs themselves do have the capability to install other versions, but is regulated by one file (/Sources/ei.cfg) that tells it what version to install. Thus you could get an all-in-one home premium disk that could install Professional (By deleting ei.cfg), but modifying files on dvds is pretty hard, and the hassle of creating a bootable dvd... but it's doable.
Also see Kravlin's comment :)