Are you allowed to use your own weapons in the military?

I have a dark beard. i was in the military and shaved twice a day during boot camp. for the rest of my career once a day was enough.

No, anyone who served is a veteran.

Is it silly to wear an NFL team jersey if you are not on the team? It is just a way of showing support.

Not in most countries.

Use whatever the Army issues to you.

Yes, as long as you are practicing medicine in the service.

there is no "law" against civilians rendering a hand salute. some already do,i.e. Boy and Girl Scouts, Police, Firefighters.
The typical or recommended salutes are guidance written into the flag code which describes flag etiquette. just as there is no law that says the flag shall never touch the ground, it is etiquette and respect that guides us.

No. most countries organized armed forces will issue the weapons they want you to use. they can better maintain and support them rather than having 100s of different types of weapons.
However you may be able to volunteer to carry one of the many squad/platoon weapons provided.

you probably wont pass the recruiting physical.