PaulMK

You could also stay on the Israeli side of Jerusalem, it's a tad bit easier because you won't be caught up in their conflict between Palestine and Israel. It's expensive to live there, but again, no problems with crossing boarders and checkpoints. There is also a nice doctor in West Jerusalem, it's called Wolfson Medical, you can go there to do your medical exam.

As Rikko said, keep refreshing the page, appointments come up randomly. Keep in mind, you need to have a medical exam two weeks before the interview, and you should stay about 2-3 weeks after the interview as well. Stay on the Jerusalem side when picking a doctor. 
 
You can set an appointment before they physically receive your files. It takes them about a month for the physical files to show up in Jerusalem, and they will email you when it does, but again, make the appointment before that confirmation.

As Rikko said, keep refreshing the page, appointments come up randomly. Keep in mind, you need to have a medical exam two weeks before the interview, and you should stay about 2-3 weeks after the interview as well. Stay on the Jerusalem side when picking a doctor. 
 
You can set an appointment before they physically receive your files. It takes them about a month for the physical files to show up in Jerusalem, and they will email you when it does, but again, make the appointment before that confirmation.

It has been two months, how has it been for you?
My Fiance and I are in a very similar situation, so we asked Jerusalem if we could transfer our case to them from Warsaw, and they are telling us they can accept our case too.
Any other problems you have ran into?

We got our case approved today. To answer the question of receiving a Russian Police Certificate digitally outside of Russia, this document seemed to be good enough for the U.S. Embassy in Israel. Our document was stamped and signed electronically, and printed out by my Fiance and I.

We got our case approved today. To answer the question of receiving a Russian Police Certificate digitally outside of Russia, this document seemed to be good enough for the U.S. Embassy in Israel. Our document was stamped and signed electronically, and printed out by my Fiance and I.