Why did Eliezer suddenly run towards Rivkah?
It was a sign of Berachah for her children, something that occurred later in the desert, when the water of the well rose by itself in their honor (See Bamidbar 21:17).
How is this indicated in the verse? Refer to 24:17:1.1.
Seeing as the water in the well rose to meet Rivkah", why in Pasuk 20, did she have to draw for all the camels? Why did the water not continue to rise by itself?
Rav Chavel on the Ramban (citing the Kedushas Levi,) and Oznayim la'Torah: The water rose the first time when Rivkah drew for herself, 1 but not the second time, when it was in order to perform the Mitzvah of Chesed, which she had to do herself. 2
Oznayim la'Torah: Because she was a Tzadekes.
Others add that the second time was a test on her Midah of Chesed, which would have been meaningless had she not drawn the water personally; alternatively, in order to receive more reward, as the Mishnah writes in Avos (end of Chap. 5) - 'L'fum Tza'ara Agra! (The reward is according to the effort!)' - Oznayim la'Torah.
Eliezer said that he will ask her to tilt her jug, and he will drink. Why did he request now only "Hagmi'ini" (merely one swallow)?
Ohr ha'Chayim: He saw that the water rose for her - a sure sign that she is from blessed seed, like Avraham. He feared lest he lose her through his stipulation, so he minimized his request, to facilitate that she will fulfill it.
Ha'amek Davar (to 24: 14,17): Most girls bend down and put the jug in the spring. It cannot be filled, so to drink, one must tilt it. She entered the spring and filled it (24:16), so he could drink a little while it is on her shoulder, without tilting it. 1
Ha'amek Davar (to 24:18): However, she in fact took down her jug so he could drink as much as he wanted.
QUESTIONS ON RASHI
Rashi writes: "The servant ran towards her - because he saw that the water rose up towards her." What indicates this in the verse?
Rashi extrapolates it, from the fact that the Torah mentions that she "filled her bucket," but not that she drew water.
Mizrachi: In the preceding verse (24:16), we are told that Rivkah had already gone up from the spring, where Eliezer had been standing (24:12). If so, the verse should have said, 'the servant ran after her.' Rather the phrase must be split into two, 'The servant ran [after her, for he saw that the waters had risen up] towards her.'
Gur Aryeh: Rashi was troubled, not by the term "towards her," but by why Eliezer ran specifically towards Rivkah. There must have been many girls at the well; why was she special? It must be because the waters rose to her. This particular miracle happened for a number of righteous people. 1 On a deeper level, a Tzadik generates blessing, and a well-spring is a source of blessing.
Gur Aryeh: For example, Avraham (Rashi to 21:30), Yaakov (Rashi to 50:3), Moshe (Rashi to Shemos 2:20).