A recent development - maybe 50 or 60 years, you can be counted be counted as Jewish if yoyr father is Jewish in the reform or liberal communities. However orthodox communities will never accept you as one, nor will Israelis. Of course, secular Jews really don't care.

Yes Trump's daughter and grandchildren will be Jewish. Americans marry out a lot so the rules are not as strict as i. Some other countries.

No mattee what yout DNA is, it does not count whether you are considered Jewish or not. I brought that up because there are a lot of accusations that Jews stole the land but DNA shows that Jews originate from the Middle-East. The claim that all Jews come from Europe and are white is disproven by DNA studies.

Ih there is a continuous line of the mothers and grandmothers being Jewish, and if it can be proven, then the children are considered Jewish even if they grow up in a secular home.

So if your great grandmother was Jewish, both male and female children would be considered Jewish. If the male married out, and his wife did not convert, then their children would not be Jewish. However if the daughter married out, the children would still be considered Jewish.

Nobody checks DNA.

I know about it because I read widely.

Jews were expelled from Arab lands as well. Very unhappy situation. A lot of people got caught in the crossfire without being responsible for the situation.