I know you posted this in March, but I wanted to answer question regarding "why did they give, ,Poland" as their birth place? "
Most immigrants wrote the placename, admin.division, country, etc. that they last knew it as, when they left there! Depending on when and where the document was created, there could be 5 or 10 years difference in time! And in the case of Ukraine, that could include multiple changes!
Here's a short interesting article that might help..
"Did Your Baba Come from Austria?" by Orysia Tracz
My grandfather was a twin, and they left Volhynia at different times and ended up separated! I have one twin writing his birthplace as one name and the other twin appearing to be born elsewhere!
So part of doing family geneology, is to first become familiar with the time period of the area you're going to research. Especially in the time frame your ancestors lived there! (that's becoming familiar with both historical and commonly called place names)
Because although one of my ancestrial lines were Orthodox clergy, they seemed to have lived in or next to villages (aka selo in Ukrainian) but when researching, I happened to find more information using Jewish resources and they call a selo, a shetl!
I hope that if this doesn't help you, then maybe it'll help someone else!