Does my DVR have to be left on for recordings?

I just got a HR24/500 DirecTV and service.

One question is do I have to leave the receiver on all the time for it to record scheduled shows? or can I turn it off and it will do whatever it needs to do the recording?

Secondly, do both inputs in the back need to be hooked up to record 2 shows at the same time? If so, can I just use a splitter? (I live in a multi-dwelling condo, so there is no second line etc, just one coax jack behind TV).

Thanks in advance!