Can I legally purchase/posses a firearm in California?

No you cannot. Expungements have no impact on gun rights. Your conviction is a non-reducible felony. No guns...ever.

Understanding that truth is a defense, what exactly are your damages?

Can I leave a shotgun and ammo in a hotel room when I am not there?

Always best to store firearms and ammo in a secure manner. Inside a motel/hotel room is not very secure and a terrible idea.

Is it illegal to panhandle for money on the streets??

Have you ever seen anybody arrested for panhandling in LA?

I stayed in jail an extra 5 days can I get paid?

Maybe. Are you sure the release date stated is accurate?

What does it mean that they opened an investigation?

Sounds like perhaps a Juvenile Court/Probation matter. Are you a minor?

Is there any chance at winning or dropping charges for a trial on battery?

Any of those possibilities are possible.

Speeding ticket dismissed with lawyer in court July '19, but Failure To Appear Notice Ret/No New Address added in November '19

There may be a mistake. Take your paperwork and go see the clerk at the traffic court.

Should I press charges on someone who assaulted me?

You may report whatever happened to the police.

Requesting All discovery from District Attorney on a resolved case?

I don’t think the DA will give you, nor anyone else, anything at this point.