Slightly more broken down processes would be greatly appreciated.
For you it is like this:
1. Contact family court at regional court (assume it’ll be the one nearest where your girlfriend lives) and make a booking there.
2. Wait anything from a couple of weeks to a couple of months.
3. While waiting, contact US embassy to arrange letter from them regarding their inability to issue certificate of no impediment. Ask their advice with regard to the whole process.
4. Also while waiting, have your birth certificate translated by a sworn translator: you’ll be needing that.
5. Go to family court and swear an oath that you aren’t married and that you are unable to get a certificate of no impediment from your government.
6. Wait a couple of weeks for the court to make its ruling.
7. Go back to court and get the waiver of need for of a certificate of no impediment. I’m not sure if you’d need to do this in person or not. If you don’t want to do it, you’d need to make that clear at the first hearing.
8. Contact Urzad Stanu Cywilnego (registry office), check the court paperwork is good with them. Ask what other documentation they will want to see and the forms thereof.
9. Go to Urzad Stanu Cywilnego (registry office), present your documentation, check they will accept all of it, book a date to get married.
10. Go to Gmina offices to get a maldunek (just in case you get a registrar who wants to confirm that you are completely legal in Poland).
11. Go to registry office and get married (check in advance what documents they will want you to bring on the day).
For your girlfriend this is the harder way:
1. Get a certificate of no impediment sent from Mongolia.
2. Have a sworn translator translate it.
3. Contact USC to make sure that the translation of that document will be acceptable and ask what other documentation will be acceptable.
4. Have her birth certificate translated by a sworn translator: if she already has one done, check with the USC that they will accept it (it may have expired, I once had a translation of my birth certificate rejected because the translation was done more than a year ago). Ask what other documentation they will want to see and the forms thereof.
5. As above for step nine for you.
And the easier way is this:
1. Get certificate of no impediment from the Mongolian embassy (assuming they will issue one), in Polish if possible.
2. Check with the USC that they will accept it (wording, stamps etc). Ask what other documentation they will want to see and the forms thereof.
3. As above for step nine for you.
Oh and the above assumes that neither of you have ever been divorced. If you have, that divorce will have to be recognised by the Polish court, and that is where things get complicated.
Still so sure you want to get married in Poland? I hear Denmark is a quick and easy place to get married; once you are legally hitched there, you can just come here for the ceremonial part.