At the very least, you should be able to record 7 shows simultaneously with your 7 tuners.
The exact number depends on the type of tuners, and the platform you're running on. With DVB tuners here in NZ, I've successfully recording 16 channels with a dual tuner device.
No. The CPU/IO requirements are generally fairly low, and should be fine with any normal computer.
You might find you have a problem recording that many channels at the same time if you got a really slow drive, or processor (like a $35 RPi recording to a MicroSD drive)