What is "b'His'atef"?
Rashi citing Targum Yonasan: B'Ishtalhayos (fainting). The same applies to "be'Atef Olel v'Yonek" (Eichah 2:11).
Radak: This refers to suffering that makes one lose the will to live. The same applies to "Nafsham Bahem Tis'atef" (Tehilim 107:5). When I was afflicted, when the fish first swallowed me, I considered myself dead.
Malbim: He was close to death due to being crowded with the fish's offspring in its womb.
How did he remember Hashem?
Radak: He prayed to Him.
Malbim: He prayed, repented and promised to fulfill Hashem's word.
How did he know that his Tefilah came to Him?
Radak: He infers this, for he remained alive.
Did his Tefilah go to the Heichal?
Radak: Shamayim is Heichal Kodsho. The same applies to "va'Shem b'Heichal Kodsho" (Chabakuk 2:20). "Hashem ba'Shamayim Hechin Kis'o" (Tehilim 103:19), "va'Tavo Sefilasam li'M'on Kodsho la'Shamayim" (Divrei ha'Yamim II, 30:27).
Malbim: Yes. Chazal said that he was in the Tehom under Har Tziyon, on which the Beis ha'Mikdash stands.