2008 Dodge Charger Dash lights

If I understand right, the "Battery" light was on when the engine was not running. It had better be. That means the alternator is not recharging the battery, and that is to be expected because it has to be spinning to do its thing. The belt on the running engine spins the alternator to recharge the battery. As long as that light goes off when the engine is running, it's working normally.

I'm not sure what the other light is. The lightning bolt normally relates to something electrical but they wouldn't put two lights on there for the same thing.

For being totally dead, there is most likely a real easy fix, and it is pretty common on all car brands. Follow the smaller battery positive wire to where it bolts to the under-hood fuse box. Make sure that connection is clean and tight. If you know how to use jumper cables, you have the skills necessary to tighten that nut, but be careful that whatever tool you use, it doesn't touch that nut and anything metal on the body or engine at the same time. Doing so will make one seriously horrendous spark and will weld the tool in place and turn it red hot in a few seconds.

What you're describing is how they're supposed to work. Various lights turn on before you start the engine as a bulb test and while the computers are performing some self tests. Your Air Bag light does the same thing.

There could be diagnostic fault codes stored in some of the computers due to the intermittent connection on the battery cable. Once that is taken care of and the problem no long occurs, those fault codes typically erase automatically by themselves. If there's a safety problem, the computer for the affected system will turn it off and turn the warning light on to tell you.

Yup. The battery would only last about half an hour on its own. It will be recharged as you drive. If the charging system didn't do its job, the battery would run down very quickly and the engine would stall. Before that happened you'd see the "Battery" warning light on the dash, the radio would get weak, the power windows and wipers would slow down, etc.