I understand that you want to know if the Slingbox allows multiple connections at the same time and if it has recording capabilities.
Although you are able to get the connection to your Slingbox from many devices (supported computers, tablets and Mobile phones), the Slingbox will allow only one connection at the time, due to the broadcast restrictions.
However, the Slingbox PRO-HD will allow you to split the TV signal if you use its internal tuner. You must connect a splitter to the cable that comes from the wall (coaxial). Then, send two lines out from the splitter: one will feed your TV set or cable box and the second line will feed the Slingbox coaxial input. The only disadvantage is that you will only get the basic cable service channels, since there is no connection through the set top box.
Regarding your last question, unfortunately the Slingbox does not have recording capabilities. This means that you will not be able record with the Slingbox. Your only option in this case is to hook up a supported DVR unit. This way, the Slingbox will allow you to control your DVR features.