The answer you get depends on the question you ask. So if you for example ask about your ancestors and say they married in Roman-Catholic church and do not mention their names (or you do but the clerk in the archives does not read thoroughly enough) then they answer they do not have Roman-Catholic records for this area and advise you to contact Rzeszow archives. In fact, Rzeszow archives does not have any Greek-Catholic records as far as I know. Generally, Most of Greek-Catholic records are in Przemysl archive and some are in Sanok archives (which is a branch of Rzeszow). In your case, they are propobly right saying they do nto have records you are interested at the moment (they may only have older records from Kuryłówka, you may be interested at some point but you will need to find the bridge) but, in my opinion, the only sense in contacting Rzeszow archives is asking them if they know where the records could be but they do not have Greek-Catholic records.
As for the civil registry office (USC) in Kuryłówka their address is:
Urząd Stanu Cywilnego
Urząd Gminy Kuryłówka
phone: tel. 0048 17 243 80 10 (this is their main number, you have to ask for USC)
I suggest to write them in Polish. You may start with writing them an email or calling and if it does not work then writing a regular letter. According to the law USC can only provide transcripts of the records to direct descendants (they charge 33zl - ~$10) and you have to give them at least a year and they do not search the records. In practice they sometimes do search and provide copies, it depends on the clerk. Generally, the smaller the place the more helpful are clerks.