I played Gravity Crash so you can play a downloaded PSN game fine. IF you mean can I download a new one play it I don't know, I'm not going to buy a new game right now.
About trophies, I can't check if ones I earned during downtime are fine because I couldn't play during the downtime. IF you are asking if the most recent ones I earn before the downtime are ok then yes they are and the reason people might no be seeing them if because they were moved to the bottom of the trophy list. Im checking the time stamp on THOSE trophies right now....and those are fine. It think I played white knight chronicles last because it think I tried to play it when the clock was reset, but I see no ill effects or miss time stamps.
edit: FLOW is not free...yet