Because only the current live buffer is saved. The previous one is lost.
Maybe. By default, HDHomeRun tuners stream direct to the client. However, all web-based streams go through the DVR server before they are sent to the client device. (And, if you have enabled "Tuner sharing" on the client device, then the HDHomeRun stream goes through the DVR server, too.)
The same situation applies as above. If it is a web stream (TVE, Locast, etc.), then a second stream is not opened. Instead, the current stream being saved on the DVR server is sent both to the client and to the file to be saved.
For HDHomeRun tuners, it depends. If you have "Tuner sharing" enabled, then no second tuners/streams are used. In that case, the stream that is being saved for the recording is also sent to the requesting client. However, if you have not enabled "Tuner sharing", then the client will contact the tuner directly and ask for a second stream of that channel to be opened.
Again, it depends. For web streams, no. Since all of the content goes through the DVR first, there is no need to open a second stream. But, for physical tuners without "Tuner sharing" enabled, then yes. A second stream will be opened, because the first/live stream is going directly to the client; the second stream needs to be opened to be sent to the DVR for recording. However, if "Tuner sharing" is enabled, then it functions the same as for web streams, and a second stream will not be started.
However, when you start recording a while watching live, the recording will only begin from the point in time that the recording was requested. This is because the live TV buffer resides on the client device, but the recording must happen on the server. (Yes, you might think that if the server is sending the streams to the clients in the case of web streams and shared tuners, it has a copy of the data, and the previous buffer should be saved. But, the server only retains a brief buffer of a few seconds of the streams before sending it out to the clients.)