Karen,
did you read the article:
Poles and Exotic Beauty Syndrome
http://www.bellaonline.com/articles/art33479.aspit is not usual in Poland to date several women while being married to one. This happen unfortunately in every culture. But you are right to be concerned in this aspect. If this man is travelling abroad a lot - here is probably the problem. He may be used to flirt and have temporary relationships with many women withough any real commitment.
Frankly - I would say, if he has a photo of his daughter and his wife in his purse - this is actually a positive aspect of him but you are right to be concerned that this may be a sign that he is not really divorced.
The easiest thing to check - do you know any of his friends? Maybe you can check whether he is divorced or married even through the Polish embassy. He cannot marry another women without being divorced and divorce takes a long time in Poland.
***I feel quite difficult to accept his daughter from my Chinese culture. ***
this suprises me. There are so many Chinese girls adopted in the USA and other countries why it would be difficult for you to accept his daughter?