Lesbian mom here. We used frozen donor sperm.
We tried for 8 cycles total out of a 10 month period. (From the research I did, it appears that 8 cycles using frozen sperm of good quality is average in women with no other fertility issues...)
We also did injectible fertility meds and acupuncture and traditional chinese medicine because I had very high FSH, which is an indication of low ovarian reserve.
I had a chemical pg on the first cycle (which was unmedicated) - possibly due to my short luteal phase which I had not identified yet. (Later took progesterone suppositories to help with that.) Loved that donor, but the bank ran out of him so we had to switch donors.
We used buckets of the next guy (who was ID release, which was at the time important to us) - but he apparently was not a good match with my eggs.
We became successful using our third donor, who we also thought was a great fit for what we were looking for. Now have an amazing, healthy, joyous daughter as a result.
In terms of what we looked for, we based the physical characteristics on my partner. At first we were looking for donors of her family's national origin, but there were too few to pick from (first donor was a fit there). Most important, however, were the availability of a photo (we want to be able to show it to our daughter at a later date), clean mental and physical health family history, and personality traits (looking for someone more easygoing and extroverted than I - the birth mother - am by nature) - and, of course, has the donor been able to get anyone pg?
Essentially, it is common to start out with very specific requirements for a donor, and, over time, become more open to someone who is successful at producing babies. Even though they are all tested for motility and sperm count, they are not all equally effective. For instance, ID release was initially very important to us, but those donors are few and far between, so we had to let go of that finally and settle for a photo.
I'll check back later to see if you have any questions.