We can't put the LAG on a set; I think he'd raise UTG with a pair, trying to thin the herd and play heads up with a legit -MIDDLE PAIR - hand. He limps with his drawing hand, trying to trap the field with a gin flop, or, heads up with a raiser that he can close PF action on. Further, I think LAG would smooth call a set, trying to trap a raiser with overs. Lag can't really put you on hearts, raising $60. You've got overs, and he knows it. So I think LAG is drawing.
A TAG button probably would dump almost any hand that directly connects with this flop after you raised. A TAG button might smooth call with QQ, hoping to push you out with a non-ace flop and going head to head with the LAG.
That's my thinking, but it's (like) Monday.
EDIT, oh, and I also thought about the implications of the heart in our hand. I think the interest in this flop tells us that a lot of hearts may lie in hands and we're relatively safe from LAG's heart-connected draw.
LAG can't have 55 here? That would fit really well. He's not gonna slowplay a set on that board becuase it's so dangerous and he wouldn't want to let 3 people see a turn card. This isn't a limit game and it's really uncommon for anyone to "take one off" in a raised pot with overcards, especially when it'd be unlikely that either overcard is good enough to win you the hand.
The TAG button's range is largely made up of middle pairs, like JJ, TT and 99.
You also keep talking about the LAG having suited connectors. What about 78hh, QJhh, T9xx. We are in terrible shape against these ranges.
If I'm reading correctly, your edit basically says that they both have heart draws and are therefore blocking each other? That seems waaay too optimistic if you ask me.