Hi Romain,
There is Przyszów – a large village in the region of Podkarpacie. I was unable to find Przeszów and Hnatkowice.
I’ll try to answer your questions:
Birth, death and wedding records, for the last 100 years or more, are kept in local village communal offices (Urząd Gminny). Copies of these same records and more are usually kept in local parish (church) offices.
There are regional archives in Poland and there is Archiwum Główne Akt Dawnych (Central) in Warsaw, but I would suggest that you start first on the very local level.
The village communal offices are obliged to issue, with a fee, official documents that are based on the records in their possession. It is up to them to give you a personal access to the record books. There is this same situation with the local priests. They are usually helpful but not always.
Columns in record books are named in Latin. First names and personal status like nobleman, labourer, have usually Latin form. All other notes are usually in Polish.
Try to arrange your research through BIBLIOTEKA GENEALOGII POLAKÓW. Through the owners of this forum.
For digitalisation of the records from Przyszów and that other place you may wait too long. There are some Polish records digitalised by genealogy.com (Mormons in the USA), but they are from the Central Archive of Ukraine in Lwów.
Regards,
Sambor