The low battery on a TPS on a cold morning, while not 'normal', is not rare either. As the tire warms, the TPS warning will go away which is why it stopped complaining at a light. I am surprised it's happening on an '11 bike, though. Quite a few of these sensors have had issues with this and have been replaced under warranty. In fact, my bike is going in tomorrow for service and that's one of the things (both of them) getting addressed.
As for the F1 no start, I don't think that's the dreaded ignition switch issue as you see something on the screen. This is something completely different, I think. So that is a new one. We've seen the F1, which is saying you have some form of fault, on the '10s upward, but it's been momentary and occurs while riding. No problem to the bike or rider.
Please keep us informed of the outcome.