Did they call to the king himself?
Radak: They called to whoever was on the wall or in Sede Choves, that they should tell the king to come to speak to them. The king commanded Elyakim, Shevna and Yo'ach to go out to them.
What is the meaning of "Al ha'Bayis"?
Malbim (Yeshayah 36:3): He was over Beis ha'Melech.
Here, and in Yeshayah 36:3, it says that Elyakim was "Al ha'Bayis." However, in the Parshah about Shevna (Yeshayah 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!
Radak: Shevna rebelled only when Ravshake returned 1 . Seder Olam 23 says that it was when Ravshake went to Kush; Shechem and his entourage went there. Sanhedrin 26a says [that he rebelled] when Sancheriv besieged Yerushalayim 2 . Perhaps it agrees with Seder Olam that he rebelled earlier, just he concealed this until the siege, so now, he tore his clothes. Perhaps the Navi told Shevna's intent to Chizkiyah, so he demoted him, so he is called only Sofer.
Radak: Do not say that before this episode, Shevna rebelled and Chizkiyah deposed him. Chizkiyah sent him with Elyakim and Yo'ach to hear Ravshake, and all three told him not to speak in Leshon ha'Kodesh, and tore their clothes (verses 26, 37)!
Radak (from Sanhedrin 26a): Shevna wrote a note and sent it via an arrow to Ashur's camp. 'Shevna and his followers surrender to you. Chizkiyah and his followers do not.' When he left Yerushalayim with his camp to make Shalom, Gavri'el closed the city gates in front of his camp.
What is "Mazkir"?
Rashi: He tells the king which judgment [Malbim Yeshayah 36:3 - or request] came first, so the king will deal with it first.
Rashi (Yeshayah 36:3): He writes the chronicles.