1)
What is the meaning of "Al ha'Bayis"?
1.
Malbim: He was over Beis ha'Melech.
2)
Here, and in Melachim II, 18:18, it says that Elyakim was "Al ha'Bayis." However, in the Parshah about Shevna (above, 22:15) it says "ha'Sochen ha'Zeh Al Shevna Asher Al ha'Bayis"! The prophecy was that he will be deposed, and Elyakim will oversee the Bayis!
1.
Refer to Melachim II, 18:18:3:1 and the notes there.
3)
What is "Mazkir"?
1.
Rashi #1: He writes the chronicles.
2.
Rashi #2: He tells the king which judgment [Malbim - or request] came first, so the king will deal with it first.