Yep, Jordan is very safe. You can walk around anywhere in Amman, at any time, and the only reason someone would approach you would be to ask if you needed assistance, and they would mean it. I've had a car drive in front of my taxi to guide it, halfway across Amman, when the taxi driver was unclear of where I wanted to go, and the car driver didn't even accept compensation from me...

Shorts are OK, but expect a few odd looks outside Aqaba and the tourist sites. Not because they are immodest, but because they are kind of seen as children's clothes or sports clothes.

Apart from that, T-shirt will work just fun (but watch out for the sun).

If you enter a mosque, you should of course cover up as a sign of respect, just like you would in a cathedral.