The timing for Just Appeals actually seems to be a fraction of a second AFTER the beat. If you think you're hitting the beat but you keep missing the JA, you're probably doing it too early. The game is probably designed this way to compensate for upscaler lag on typical HDTVs...
Your idol's tension level makes a huge difference in auditions. If she's in high tension (any red in the bar) you get 15% (!) more points per appeal and the memory roulette reel has 75%/25% good/bad panels. Mid tension (yellow/blue bar) and the memory reel is 50%/50% good/bad. Low tension (blue/black) and you get 10% less points per appeal, and the memory reel is 75% bad! Basically, you should only even think of taking on 1-winner auditions (Specials, 70K) in high tension. If your idol has dropped to mid tension or only has a sliver of red left (risking dropping to mid if you flub the random pre-audition communication), do an "easy" audition to get her tension back up.
Oh yeah, don't worry about "saving" your memories for the ending. The ending uses the total number of memories you've accumulated over the entire game (including ones you used in auditions), not the number you have remaining.
Weird tip: If you have a 4x3 TV or monitor, you can hold X while the game is booting (before the Bandai Namco logo appears) to force the game to display in widescreen (i.e. letterboxed). This makes the memory roulette easier since you can see more of the reel, at the cost of making everything smaller (I bet it looks pretty shitty on a SDTV, but it's nice on a VGA monitor)