We need shallow DOF because of video screen behind the speaker so standard prosumer video cams look bad.
You need to think about focal length, you need to think about perspective (perhaps more distance between the speaker and the screen) the larger sensor will help a little, but if its a problem with a conventional camera I don't think its something that a DSLR in and of itself would enitrely eradicate.
2nd problem with DSLRs and video screens in shot is the scanning pattern. If you use a camcorder off the same phase mains then it usually isn't an issue, if you use a camcorder or camera off of battery it can be, and can be compounded if the video screen is traditional interlace and the camcorder is progressive (as on the EOS cameras) canon video cameras sometimes feature a clearscan mode to eradicate the scanning pattern, I'm not sure if the 1DC or c300 etc feature this, I'm fairly certain that it would be a bit specialist for the lower EOS cameras.
1: Max recording time on any card depends on the size of the card. Max file size under the FAT32 formatting is 4GB. This equates to around 11 mins per clip at 1080. More recent EOS cameras such as the 1DC, 1DX, 5D3 and (I think) the 650D feature continuous recording across files up to a clip maximum of 29mins (this is to avoid a different product classification) If you load magic lantern onto some cams (5D2, 60D, 600D, 550D) you can lower the recording data rate to get longer run times within the 4GB cap.
2: No. Big cards tend to be very very expensive, and there is an arguement that if you have smaller cards and one fails or corrupts it's less of a problem than if a huge card fails.
3: Yep you can start a new file straight away, prolonged recording times may cause the sensor to overheat.
4: For this application I would say you are better with conventional camcorders. Maybe Canon X300s or x105s or similar.