In answer to your questions as a fellow Kat player.
1. I find that it's situational. AP is very much a one-trick-pony. Once your ult is down, you're basically useless for the rest of the fight. If you're facing a team with very little CC, then definitely run AP. Your rylai's slow will ensure that DL is a triple kill or so every time you use it. AD, however, is better in most situations not because of damage capabilities but because after your DL is down you can still slap things pretty hard with BB and melee. Good teams will be interrupting your DL as soon as you cast it, so AD build gives you some leftover damage.
2. Yeah, in that situation.. don't play katarina, lol. I found Amumu because I hate being on tankless teams.
3. Once you have a couple points in Q (I go Q-E-W-Q-Q-R, though that's unconventional) it does some **** good damage as a harass. Use shunpo for last hits or closing the distance for a kill, use your Q for primarily harassment and zoning unless you want to get the last hit on a few creeps at the same time.
4. Defensive items are always a good idea no matter what character you're playing. Banshee's is nice simply because of the passive effect and the magic resist. It all depends on who you're up against, though.
As far as Frozen Mallet goes, no, it's not really a good choice. It doesn't get activated by your ulti because physical damage doesn't matter. FM is only activated on melee autoattacks.