Rabbi Guggenheimer in "Reading the Torah", parsha Toledot on Bereshis 26 suggests a very simple explanation:
The Avimelekh to whom Yitzchak goes in the famine, in contrast to Avraham's king of Gerar, is not Canaanite but Philistine ("invader") even though he and his army chief retained the Canaanite titles from people which they displaced. This means that one has to fix time of Isaac shortly after the big earthquake which destroyed Knossos, the capital of Crete, and forced the Sea People to invade the Eastern shore of Mediterranean. The example here shows that the invaders at first took over the names and institutions of the people which they displaced. By the time of Yehoshua they had no kings any more but rulers סרנים, Greek τυραννοι - tyrants. The Avimelekh here could not possibly have learned from experiences of Avimelekh of Avraham's time, and as a Greek style king could not take citizen's property (including wives) at will.
Bereshis 26:15. Yitzchak returns to Beer Sheva. The Philistines had destroyed all the cisterns in the foothills. Which person in his senses would destroy water sources in an arid area? It seems the Philistines deliberately created an empty space between themselves and the Canaanites, as a defensive glacis. They did not give up claim to the surrounding land and therefore laid claim to the water. The permanent trouble with the Philistines started only with Yehoshua attaking through the defense area (Yehoshua 10:41).
Bereshis 26:26-31. Yitzchak in Beer Sheva receives Avimelekh and his army chief as a friendly sovereign. He swears not to have hostile relations with the Philistines but does not pay tribute to this Avimelekh either for land or for water. The political situation is totally different from that of Abraham's time. (By the way - this is the second naming of the place Beer Sheva, why the need?)
However, his words go against Gemara Chullin 60b (thank you @Yø-c Ro for helpful discussion):
The verse states: “The five lords of the Philistines: The Gazite, and the Ashdodite, the Ashkelonite, the Gittite, and the Ekronite; also the Avvim” (Joshua 13:3). The verse is difficult, since it first said there are five lords of the Philistines, but it then lists six. Rabbi Yonatan said: There were in fact six lords, but the greatest of them were only five. Rav Ḥisda said to Rav Taḥlifa bar Avina: Write this statement about the greatest [arunekei] in your book of aggada, and explain that word. The Gemara notes: And this statement disagrees with the opinion of Rav, as Rav says: The Avvim were not Philistines; rather, they came from Teiman.
This is also taught in a baraita: The Avvim came from Teiman. And why were they called Avvim and not Teimanim? Because they corrupted [ivvetu] and destroyed their place of origin when they left. Alternatively, they were called Avvim since they desired [ivvu] many deities. Alternatively, they were called Avvim since they were so fearsome that all who saw them were seized by convulsions [avit]. Rav Yosef said: And each one of them has sixteen rows of teeth.
The Gemara cites another statement related to the Avvim: Rabbi Shimon ben Lakish says: There are many verses that are seemingly fit to be burned as books of the heretics, since they appear redundant or frivolous, and yet they are themselves the essence of Torah. For example, the verse states: “And the Avvim, that dwelt in villages as far as Gaza, the Caphtorim, that came forth out of Caphtor, destroyed them, and dwelt in their stead” (Deuteronomy 2:23). What practical difference does this make for us?
The verse teaches the following: Since Abimelech, king of the Philistines, administered an oath to Abraham: “That you will not deal falsely with me, nor with my son, nor with my grandson” (Genesis 21:23), the Jewish people were prohibited from conquering the land of the Philistines until four generations had passed. Therefore, the Holy One, Blessed be He, said: Let the Caphtorim come and remove the land from the Avvim, who are the same as the Philistines, and let Israel come and remove it from the Caphtorim, circumventing the prohibition.
Targum Onkelos on Bereshis 26:28 and Rashi there both state that the covenant between Yitzchak and Avimelech of parshas Toldos was just an extension of the covenant of their fathers, between Avraham and Avielekh of Vayeira. Thus one can conclude that the Avvim came to Philistia by the time of Avraham and they separated from Canaanim with the defensive glacis as R'Guggenheimer explains to prevent invasions at the time of drought!
Based on this clarification, one can answer the questions:
The wells were important for people to survive during the famine.
The Torah shows that Yitzchak did not give up and gave his best effort to survive in Eretz Yisrael without descending to Mitzraim, like Avraham did.
The Philistines stopped up the wells which had fallen in the defensive zone to separate themselves from Canaanites.
The Philistines contested that Yitzchak was using their water.
In the arid area it is really difficult to find new wells of water, especially if it is a desert. Yitzchak tried to rely on Avraham's findings, whose servant Eliezer was master of finding where to dig for water.
Yitzchak gave the wells the same names as his father did out of respect and grattitude for his father, whose efforts helped him and his household survive. In an arid area, each well could give rise to a settlement, and that's why it was getting a name.
