1. I'd still stick to the street. Study after study shows it's still the safest place to ride. As for the helmet, you are fine without one. Many say you need one, but the science just does not back that statement. ATM the jury is still out on the helmet. Some studies claim a huge benefit, others none, and yet others a harm to wearing a helmet. So without that scientific backing, it really comes down to personal choice. Hence the reason I don't and will never wear one.
2. All I can say is get a better bike. You will have to stop at lights whether you ride in the street or on the sidewalk. Best to just be prepared.
3. is that a top rack mounted backet? If so I have no suggestion as I find them silly. I stick with large dutch style panniers. No problems then getting on or off.