Thanks for posting your query at DoctorSpring.com
Let me clarify your doubts -
1. Yes the ultrasound on july 3rd is the gestational age. It is calculated from fetal size, but it is the gestational age as most algorithms calculate the gestational age directly from the size of the fetus.
2. Based on the scan, it is very probable that you conceived in the second week of May, I would say after 10th of May, more probably around the 20th - 22nd of May. Blood tests cannot be used to date a pregnancy, they are not reliable.
3. Negele formula and EDD based on scan are 2 different things. Negele formula is for a woman with 28 day cycles. For a woman with 35 day cycles, we just add 7 more days to the EDD. So if your LMP was 24th April, your EDD would ideally be 31st January 2016, and 7th of february , 2016, by adding 7 more days for a 35 day cycle.
Going by your ultrasound scan, your EDD would be 14th of February ( LMP - 7th of May, and then 9 months and 7 days )
4. I would say you conceived around the 22nd of May, and your due date would be 14th of February, as in case of delayed cycles, the first ultrasound scan is the most reliable indicator and the scan dates are considered final.
Feel free to discuss further.