Yes, they did. If they don't accept your international insurance, the consulate may not give you a visa and will ask you to send them the right insurance. In term of travel, I think things went well. It was quite easy leaving the US. All they checked was my RTD and COVID-19 vaccination card and that's all. When I arrived in Istanbul airport, it took them 3-4 minutes to check and confirm my RTD at passport check in counter. When traveling back to the US, things were relatively easy but I missed my connecting flight because the flight was delay for 25-30 minutes and you have to go through a second passport check in in the US and that took me another half an hour just waiting for my turn. The only question the officer asked me was my US address and that's all.
If you are traveling to Turkey with RTD soon, you have to make sure your RTD is valid, have COVID vaccination card/test, and have most of your USCIS approved/pending documents because I think that would make their work easy and I guess that could have been the reason they didn't ask any questions about my status. The documents that I had with me was copy of my asylum approval letter (they asked me if I had the original copy though), I-94, receipt of your pending green card application if you have applied, the address, EAD and a copy of my RTD.