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@Cousin_Isobel

Oof, Cousin, question (a) is a doozy, in part because it is of course very complicated and in part because it is very depressing. Let me return to that in a second.

(b) There are four different possible situations that a Palestinian can be born into.

Situations 1 and 2 are quite similar, so let me lump them together for purpose of your question - Palestinians born in Gaza or the West Bank cannot become Israeli citizens. At best, they *may* be able to get a work permit which does not allow them to sleep in Israel but allows them to enter between certain hours of the day for work only. These permits are highly coveted by the dispossessed population, there are not enough to go around, and their amount can be raised or lowered by Israel as a bargaining chip.
One exception is that Gazans and Palestinians from the West Bank, if they marry an Israeli citizen, are *sometimes* able to receive residency, which entitles them to some but not all rights that citizens have.

Situation 3: East Jerusalem. A Palestinian born in East Jerusalem is given a residency permit automatically, which allows them to travel and work within Israel. They are allowed to apply for citizenship, and in theory (and often right wing Israelis will falsely claim) it is supposed to be available to any East Jerusalemite. But it is actually a process that only some succeed at, as it is convoluted, expensive, and many are rejected.
East Jerusalemites who apply for citizenship are also often harassed by fellow Palestinians for normalizing the occupation, so there is a social cost as well. However, my (incomplete) understanding is that this is becoming more acceptable in some sectors of East Jerusalem society.
East Jerusalemites can in theory move into Israel proper, although in many cities and towns this is de facto impossible because of implicit and explicit racist policies against renting to Palestinians.

Situation 4 are Palestinian citizens of Israel, who are, well, citizens, so they formally enjoy all rights that an Israeli Jew enjoys. (Of course they face discrimination, but in theory they are legally equal.)