Hi there. I am sorry for your losses.
I am not a doctor, but I had 3 losses before having a successful pregnancy so I may be able to share some insight. I had many of the same questions that you typed above.
My RE said that usually, if a sperm is bad, it swims slower than the "good" sperm. The "good" sperm get to the egg first, thus decreasing chances that a bad sperm will fertilize it.
With that being said, smoking, drinking and

will decrease sperm count and/or quality. It takes sperm about 90 days to develop and mature, so if he quits now, the sperm will be healthier in 3 months!
For progesterone...your OB can test your progesterone at about 7 days past ovulation. I believe that as long as your post-ovulatory progesterone level is over 10, it can support a pregnancy, although most OBs will give supplements if it's under 15.
Now...I had 3 losses in 7 months. My first loss was at 10 weeks, after we had seen a heartbeat. After my 3rd loss, my OB decided to run "recurrent pregnancy loss" testing on me. (My DH had to do a high-tech sperm sample test too.) I had a hysteroscopy and had about 7 vials of blood drawn. Turned out I had a blood clotting factor that was causing miscarriages. During my 4th pregnancy, I took a baby aspirin each day and also did blood thinners. If you'd like to see how my 4th pregnancy turned out...look at the little miracle in my avatar. Today, 11-10, is his 10-month birthday.
It's crap that a doctor will make you wait to have 3 losses before allowing the "Recurrent Pregnancy Loss" testing. However, maybe since you've seen a heartbeat and then had a loss, your doc will consider it. It's worth asking for. Good luck, and let us know how it turns out. Please hang in there and don't give up hope.