What had Shlomo prayed for?
Malbim: (a) Hashem should put His Shechinah there constantly, and hear the prayers of those who pray there. Hashem accepted this (b) The persistence of Malchus Beis David.
How was Hashem Makdish the Bayis?
Malbim: (a) "Lasum Shmi Sham" - it will always be called Beis Hashem. (b) Refer to 9:3:3:3.
What are "Einai v'Libi"?
Rashi citing Targum Yonasan: My Shechinah will dwell in the house if My will is done; My eyes will be there if My heart and desire are there.
Radak: My Hashgachah and My will.
Malbim: (a) Hashem will supervise it with a special miraculous supervision. The eyes allude to this. (b) It will be with great love; the heart alludes to this. These two are only while the Mikdash stands. At a time of Churban, Hashem's name is called on it.
What is the difference between "Ad Olam" and "Kol ha'Yamim"?
Malbim: "Kol ha'Yamim" are all the days for the endurance of a matter, e.g. "Kol ha'Yamim Asher Ben Yishai Chai" (Shmuel I, 20:31), "Kol ha'Yamim Asher Atem Chayim Al ha'Adamah" (Devarim 12:1). All the days that it will be built, My eyes will be there. My name will be called on it "Ad Olam" - without ceasing, even at a time of Churban.