1) If a pregnant woman get into labor on Shabbat or Yom Tov, which melachot are permitted for her to perform?
It is permitted to perform all melachos that are required for getting her to hospital and those directly related to the birth. Concerning others accompanying her to hospital, the general custom is that only the husband accompanies her on the drive to hospital.
In principle it is also permitted to perform other melachos, where the lady in birth requires or craves something that will sooth or calm her (Shabbos 128; Shulchan Aruch 330:1). Thus, if the mother in question is adamant that she needs her mother with her, too, the mother can also come. The general rule is that one should act in the same way as one would act on a weekday (328:3)
However, this idea is based on the concept of pikuach nefesh (danger to life), and therefore should not be relied on light-headedly, Only if the lady giving birth insists that she requires something (e.g. she would request her mother’s presence even if her brother was getting married and her mother receiving guests), or where it is clear that without it she will be significantly disturbed, is it permitted to perform a melachah.
If a melachah must be performed, one should do so by means of a non-Jew or a shinui (an unusual way of doing it) where this is possible.
There are many details of this matter, and each question should be asked separately.