Just a big update about everything. We emailed embassies in every country that doesn't require a visa for Russian citizens. The first embassy that replied was the embassy in Serbia. They actually would accept transfer but they recommended to seek for other options due to their backlog (we'd have to wait a year before scheduling), embassy in Kazakhstan refused to take our case and the embassy in Jerusalem accepted our case without any problem. We sent the picture of denied Schengen visa application though. It took us about a month to transfer our case to Jerusalem - 2 weeks waiting for their response to our request and 2 weeks for the transfer itself. After that we had to wait for 3 weeks before the embassy released slots for scheduling. Then my American fiance called the clinic in Jerusalem to schedule my medical exam which took place 2 weeks before the interview at the embassy.
I recommend if you can to fly to Israel together. It'll make going through the passport control so much easier. I arrived by myself and the stall me for about 2 hours at the passport control even though most of that time I was waiting for my interrogation and for them to give me my passport back. The interrogation took about 15 minutes. If you go alone I highly recommend to print everything you need for the interview before arriving to Israel. The officer asked my to show them the invitation from the embassy, pictures of my American fiance and a copy of his passport. By the end of it they told me they'll call the embassy and that I should just wait. In about 40 minutes I got my passport back.
About medical exam - the only thing I was nervous about is COVID-19 vaccine since they specify in an email they send you that you have to have it. BUT they make exemptions for Russians. I actually lacked another necessary vaccine but they just gave it to me at the clinic. They told me to get the results at the day of my interview in the morning which concerned me a bit because the website says the clinic opens at 8AM and my interview was supposed to be at 8AM as well. But if you just want to receive the results you can go to the clinic at around 7AM.
If you're a male you have to have your Russian military documents with you. It was a surprise for some so now they specify it in the list of required documents when you receive a email from the embassy. The interview itself took about 20 minutes and I was asked questions about how we met, our plans for the future, etc. Expected questions for a couple. By the end of the interview the officer congratulated me and said they're going to approve my visa in about 10 days. And in 10 days I received an email with invitation to get my passport back with visa in it at the embassy.
The passport control in the US was pretty easy. I just gave a package I received at the embassy to the officer and he asked me only once question - Where are you going to live? And that's it.
I just want to thank everyone on this forum for their posts and help. God the process should not be that stressful. And special thanks to @rikko. Your experience and your advice helped us a lot. I hope your case will be approved soon and you can leave all that nonsense behind.