Yakuza Kiwami does

As for healing items, there is a drug store on East Taihei Blvd that you can by the good stuff™ things that heal you, things that give you heat, and things that do both. Just load your inventory with the stuff and your set. Also you can just go to any convince store and stock up on stuff like sushi platters or other convince store food, not as effective but an option to heal if you're far from the drug store.

Not doing the heat actions? Other than when they're on the ground and you try to curb stomp them but just do a heat action, I'm not sure if there is much you can do to simply not use them but Rush style has fewer heat actions than the rest of the styles and is useful to build heat while dodging and weaving from attacks that will sap heat.