Dogs are only allowed on the beach with permits, on leashes, and during certain hours. The MHWL does not apply to dogs, only humans.
I swim in Eastern Lake, but do so near the outfall - think the further away from there one gets, the higher the likelihood of meeting a gator or huge snake (which is also why I do not venture north of the bridge in a small craft).
I haven't spent any time on Eastern Lake, but a fair bit on Western Lake. While I revel at its beauty, something instinctively tells me it's for other creatures besides me.