1. The optimal way to warm up is do a light jog for 2–3 minutes. This can be done on a treadmill, stationary bike, whatever it may be.
2. Stretching is fine. Make sure you do it after you are done, known as the cool down. When you stretch before exercise you risk more injury. I would explain why but that is a whole other question to have answered.
3. You can keep the crappy shoes you have now, but make sure the sole is not worn down. If you like the crappy shoes you have, you can save money by buying an insert. If your shoes are crappy you increase your risk of injury. For example, shin splints, plantar facitis, etc.
4. Anytime time of day is good to run. If you are running outside with a lot of traffic, I have heard that running in the morning is best because there is less pollution.
5. My favorite songs to run to are songs that have a steady beat and are chill just good underground hip-hop.
6. You should eat before and after you run. You should eat a light snack about 20 minutes – 1 hour before running and then eat as soon as you can after running.
Hope this helps!