Was the oil Shemen ha'Mishchah?
Bamidbar Rabah 8:4, Rashi (Kerisus 5b DH v'Lo Malchei), Rambam (Hilchos Melachim 1 7). Yes. 1
Radak: No. That was used only for kings of Beis David (Horiyos 11b). Rather, it was balsam oil.
What is the significance that he anointed him from a flask [of oil]?
Radak (from Megilah 14a): ("My horn is raised" (2:1), but not my flask. David and Shlomo were anointed with a horn (of oil), and their kingships lasted. Sha'ul and Yehu were anointed from flasks, and their kingships did not last.
Was it proper to kiss him?
Radak citing Bereishis Rabah 70:12]: Every kissing is of Tiflus (lewdness), except for three - of greatness, of Perakim 1 and of separation. Here was of greatness; Aharon kissed Moshe by Har Sinai (Shemos 4:27), i.e. of Perakim; Orpah kissed Na'ami (Rus 1:14), i.e. of separation. R. Tanchuma adds, also of kinship, like Yakov kissed Rachel (Bereishis 29:11).
Malbim: Yes - through this, his spirit clung to him, and he drew on him a Kadosh influence that prepared him for prophecy afterwards among Bnei ha'Nevi'im (verse 6).
I.e. after being reunited after separation. Yefe To'ar - Tiflus applies only to a man kissing a woman. The Midrash gives examples from verses about two men, even though that is permitted in any case. Etz Yosef implies that we discuss men kissing men. ('Greatness' must include great Chachmah or Midos, for R. Yehoshua kissed a clever boy, Shimon ha'Tzadik kissed a Nazir, Yehoshafat is praised for kissing Chachamim... (Eruvin 53b, Nedarim 9b, Kesuvos 103b) It seems that R. Gamliel kissed R. Yehoshua (Rosh Hashanah 25a) amidst gratitude. (PF)
Why did he not anoint him in public?
Malbim citing Mahari: It was in order that he could cast a lottery afterwards [to pick the king]. Had he already anointed him, the lottery would seem superfluous, and they would suspect trickery to make the lot fall on the one he had anointed.
Malbim: Since he was anointed with a flask and balsam oil, and not with a horn and Shemen ha'Mishchah, he did not want people to realize that he was anointed only for a Nagid (leader), but not to be a king. 1
Since no king was anointed before, how would people realize that his anointment was not for kingship? (PF)
Why did he say "ha'Lo Ki Meshachacha"?
Radak: This is to strengthen the matter and make the person zealous. We find like this - "ha'Lo Shelachticha" (Shoftim 6:14), "ha'Lo Shamat Biti" (Rus 2:8).
Why did he say "you were anointed to be a Nagid", and not 'a king'?
Me'iri (Horiyos 11b): Hashem never called him a king. Only after he sinned through not waiting for Shmuel, and regarding Amalek, Hashem said that He despised him from kingship (13:13, 15:23).
Malbim: He was not promised that his children would inherit his kingship, like one who is anointed with Kodesh oil - "va'Tarem ki'R'em Karni" (Tehilim 92:11).
Rashi (Kerisus 5b) writes that he anointed him with Shemen ha'Mishchah. It is used only for kings of Beis David (Horiyos 11b)!
Rashi (5b): From David and onwards, it is forbidden for other kings. 1
Me'iri (11b) Sha'ul's kingship was 'borrowed' until there will someone in Yehudah proper for kingship. 2 It is as if Sha'ul was from Yehudah (i.e. the permanent kingship).
Kesef Mishneh (Hilchos Melachim 1 7): Sha'ul was different, for he was the first king.
Sha'ar Yosef (the Chida, 11b): Hashem told Shmuel to do so (a Hora'as Sha'ah).
Tosfos (5b): We learn from "Ki Hu Zeh" (16:12). It did not apply to Sha'ul, who was anointed before this.
David became king two years after Shmuel. Perhaps he was not yet proper for kingship, until he bore the afflictions of being pursued, and twice passed up the opportunity to kill Sha'ul. (PF)