Getting your hands on documents not identifying you as Jewish was a very important way for Jews to survive the war. While the Nazis used racial, not religious, criteria for establishing who was Jewish, church records (baptism records especially) were used to prove non-Jewish status. There was still danger in this, not only to the forger, but also if holes in their story were recognized or if they ran into someone who knew them as Jewish. Sometimes such people were sheltered by non-Jewish friends, who could support their backstory.

I am unaware of the reverse. First of all, if you somehow pretended to be Jewish in 1945 you would not move to Israel, you would be in a Displaced Persons camp. You might protest this or attempt to illegally move to Israel, but immigration to Israel was for the most part not an option for Jews in Europe until Israeli independence in 1948. British restrictions on Jewish immigration to Palestine remained, and they were the subject of a series of major international incidents and the focus of a lot of attention, which eventually caused the British to give up on Palestine completely.

Even if you did get to Israel, it was no cakewalk. Israel, like Germany, was recovering from war and was dealing with huge numbers of refugees in an acute housing crisis (a significant proportion of Israelis lived in ma'abarot, refugee camps, in the 1950s). Unlike Germany, Israel did not have large numbers of foreign soldiers enforcing political stability. Israel's economy was completely unable to cope with these challenges, and a system of peacetime rationing was in place. And, of course, someone who did this would have to deal with an unfamiliar culture and have to learn a completely unfamiliar language.

That's not to say it couldn't've happened, maybe it did. But I am unaware of a German pretending to be Jewish to move to Israel, and such a thing, though possible, would not make a lot of sense.