It seems that you have some general questions about how the Slingbox works.
First, I will explain you the main differences between the Slingbox SOLO and the Slingbox PRO-HD. The main differences are that the Slingbox PRO-HD allows you to connect up to three devices while the Slingbox SOLO allows only one connection. Another important difference is that the Slingbox PRO-HD is the only one capable to stream the HD signal.
When accessing the TV from a remote laptop through the slingbox, does anything in particular need to be happening with the TV at home? Does it need to be on?
It depends. If you connect the Slingbox PRO-HD to the same AV source your TV is attached to, you both (remote viewer and home viewer) will share the same TV signal. You can avoid this situation placing a splitter on the coaxial cable that comes from the wall. From the splitter you will send two lines: one to the Slingbox and the second line will feed your set top box. The only disadvantage is that you will only get the basic cable service channels on your Slingbox.
That means that it does not matter if the TV set is on or off.
Can someone at home be watching TV at the same time as someone watching remotely through the slingbox?
If you set the splitter as specified before, yes. However, the Slingbox user will only get the basic cable service channels. A workaround is to connect the Slingbox to a dedicated set top box (cable box or DVR). This way, the Slingbox user will be able to get all of the channels without affecting to the home viewers.
Can you watch different channels at the same time from the home TV and the remote computer?
Same answer as before.
In all case, I recommend you to check the following links in order to get all the information about the Slingbox SOLO and the Slingbox PRO-HD:
Hope this helps!