(Asker here; I have just returned from my trip to Jamaica.)
1) In the touristy spots - Montego Bay, Negril, Ocho Rios, you will generally have no problem paying in USD.
I am not sure however about the less touristy spots, because by the time I got to such spots, I already had the opportunity to change my USD into JMD.
Which is what you want to do anyway because vendors accepting USD will usually give you a bad rate. For example in Dec 2013 one could get 103.5 JMD per USD at most cambios, and sometimes even 104 JMD per USD, but when accepting USD they generally used the rate 1 USD = 100 JMD.
2) Nobody ever had any problem with any of my USD notes being old, crumpled, dirty, etc. USD coins do not seem to be accepted though.
Misc advice: Do NOT exchange your money at the MoBay airport, where the rates are nothing short of a colossal scam (something like 1 USD = 86 JMD even though the outside standard rate was 103.5).