That seems like the iPad isn't getting enough current to charge properly. Have you tried getting a replacement charging lead and official charger? I've found this seems to happen over time (I look after 300+ iPads at work), and getting a new charger and lead fixes it 95% of the time.
Are you charging using an official Apple charger? I've found other chargers, and USB port charging can often not provide enough current to Apple devices to charge them properly. They'll do as you are seeing, sit on the charging screen but not actually get anywhere.
It shouldn't do, no. All phones should still turn on without a SIM card, as you would still be able to call the emergency services without a SIM card. What is more likely to be the cause is the battery not being charged, either through the battery being faulty, or your charging lead.
I'm afraid not. USB itself isn't a video/audio port. You'd need to connect with something like HDMI, or a VGA lead and audio cable. The USB port on your TV is likely for something like direct video file/photo file playback.
If you still have this issue after trying the above, there is another possible cause—crud in the charging socket on the phone! I tried the above and other "software related" fixes, and they would only work sporadically. I then took a torch and looked into the charging socket. Inside, there was a significant build up of fluff and the like. So, using a fine scalpel blade, I removed it all. Now, the charging works perfectly every time I plug it in. Warning: The blade will need to be very fine in order to fit behind the plastic piece in the middle of the connector. If you use anything too thick, you could break the plastic piece.