yah you can, sometimes it won't say you can save a replay, but if you press the button it will still ask where you want to save the replay(this means the icon that appears near the bottom of the results will not be there, but pressing the button will still save it). (the person with the code on will have to save it, the code won't transfer to your friends wii. if they want the replay, simply send them the replay after the game.)
As for the desyncs, you don't have to take off texture hacks. you must have some code that causes a change on your screen but not ur friends. (basicly you should only have the long replay code and the file replacement codes on. no glitch fixs or anything that would affect a character.)
recommend limiting your online codes with unhacked wiis to texture hacks(they won't apply if they choose the color, but they will still apply to you), long replays, the music modifier wifi code(no trouble with that so far, not to be confused with the my music swap code. this code only plays one song.) and the wifi character id code(not for cheating of course). SOMETIMES CODES THAT AREN'T BEING USED WILL STILL CAUSE DESYNCS.(if you have super mario on and no one is mario, there is still a chance it could desync.)