What does "Ariri" mean?
Rashi: It means either without heirs (Menachem ben Saruk) or without children (Rashi, Ramban, Targum Onkelus and Targum Yonasan). 1
Seforno: It is without a son to take my place. 2
Seeing as HaSh-m had already promised Avraham that He would give the land to him and to his children (12:7; 13:15), how could he even think that he would die without children?
Ramban: Tzadikim are always afraid that their sins may cause them to lose all their merits. 1
On what basis did HaSh-m reassure Avraham that he was still due to receive abundant reward? Why were his fears (that he had lost all his reward) unfounded?
Seforno: HaSh-m was informing him that, not only was his reward not diminished, but on the contrary, it had increased, due to the Chesed that he had performed with his nephew Lot. 1
Ramban: Refer to 15:1:2:3.
Da'as Zekenim: Avraham was worried that his merits had been diminished by killing the kings. HaSh-m informed him, that, on the contrary, he had removed thorns from the vineyard, for which he would receive reward. 2
Why did Avraham refer to Eliezer as "Damesek Eliezer"?
Rashi #1 and Ramban (and most other commentaries): Because he hailed from Damascus.
Rashi #2 (citing the Midrash) and Targum Yonasan: Because, together with Avraham Avinu, he chased the kings as far as Damascus, and experienced many miracles.
Rashi #3 (citing Yoma 28b): "Damesek" is the acronym of "Doleh u'Mashkeh (mi'Toraso Shel Rabo)'' - he drew from the well of Avraham's Torah and taught others.
Seforno: He was a nonentity who was not even known by his name, only by his town.
What is the connection between the facts that Avraham had no children, and that Eliezer managed his estate?
Rashi and Seforno: He was "complaining" that his affairs were being looked after by his slave, because if he had a son, then he would be the one to sustain his household. 1
Moreover, the Seforno adds, one cannot compare a slave, who manages his master's property based on fear, to a son, who manages it based on love.
QUESTIONS ON RASHI
Rashi writes: "The steward of my household is Eliezer - ... whereas if I had a son, he would be appointed over my possessions." Did Avraham yearn for a son for such a mundane reason?
Gur Aryeh: HaSh-m had told Avraham, "your reward is very great" (15:1). Avraham wondered, "If You refer to reward in this world, Eliezer is in charge of my possessions, and I have no desire for them. If you refer to the World to Come, would I not have so much more reward if I leave a son to continue teaching Torah after I depart?" Rashi therefore quotes Chazal that Eliezer was known as "Damasek" because he would draw from and spread (Doleh u'Mashkeh) his master's Torah - and Avraham wished that a son would assume that role.
Lev Eliyahu (Bereishis p. 51-54): Surely he was not concerned about his property, rather, about continuing to teach people Darchei HaSh-m. Eliezer came from the cursed Kena'an. Even though Eliezer himself was a great Tzadik, we do not find any importance to his descendants (or those of Lot, or of Avraham's other Talmidim).