You won't get an STD from sex if you are both STD free (keep in mind that it is possible to have an STD without any symptoms so either of you could have one and not realize it!). You can get STD from non sexual activities though (like from hot tubs!)
Get tested, just play it safe. Its to easy to say your clean. For your own piece of mind, both of you get tested, prove to each other you are clean. Its not a way of accusing your partner, its a way to prove yourself.
Probably not. But diseases/viruses like Mono (its an HPV) yuo can get from breathing the same air as someone with mono, and you can take it home and give it to your spouse while the virus is active (sour throats, lots of congestion.)