Level up HP only when you feel you need to. Prioritize Sleights then CP like you said, and take HP whenever you feel like you need to lighten the pressure a bit.

Sleights aren't limited to the named combos you learn through level ups. Simply stacking three cards together is a great way to near-guarantee a successful attack without fear of your cards being broken by non-boss enemies. Sure, the first card of the stack becomes unreloadable, but that won't matter if you have a lot of good cards in your deck (hence the emphasis on upgrading CP). And besides, fights against regular mobs shouldn't last long enough to make running out of cards a real danger. Save the careful management for boss fights.

You can get new cards by whacking environmental structures like you said, and also by purchasing card packs from Moogle Shop rooms. Agrabah is a great opportunity to rack up a lot of Moogle Points. If you get an Aladdin card, use it and a bunch of Moogle Point orbs will spawn with each slash that Aladdin makes against an enemy. Stack the Aladdin card followed by two other cards to ensure it won't be broken (especially if the Aladdin card is low value).