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1. How often are you barreling this turn as a bluff, but with no heart in your hand?
Almost never double-barreling complete air without a heart. The 3h kind of neutralizes your nut advantage, though you can still press your range advantage. Your equity distribution looks something this OTT:
And here's what the solver recomends:
You can see that it's using a small sizing here. This is usually the case when you have range but not nut advantage. Regardless, you have so many good bluffing candidates that there's no need to barrel non-heart air combos.
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2. When villain takes this line on this runout. How often do they have the flush?
It depends on how wide they're flatting pre and how passively they play their draws. From my range assumptions, their betting range on the river is like 20%-25% flushes, maybe a bit more if he doesn't balance well. Ranges should still wide enough in this line that you can call down with top pair+, however it might be an underbluffed spot in practice given the massive sizing.