As stated, either side of SXM will work and since it is such a small island you will get an opportunity to see both regardless where you stay.
If you go in July there should be less congestion on the road and at the beaches as well as fewer folks at the restaurants.
No beaches to avoid and all beaches are public. Some are more accessible than others.
No bug problem to speak of. My sister lives in South Carolina and we have seen more there than in SXM.