Did most of the people that Jesus heal believe he was God? If so, why does the Bible not record them asking for salvation? 1) Most of the people that Jesus healed &mdash; did they believe Jesus was God?

2) If so, why did they ask for temporary things (let me walk; let me see; get rid of my diseases; heal my son/daughter) rather than "Tell me I'm saved"?

In this sense, the repentant thief was an absolute genius. He could have probably just as easily asked, "Please, let it be that after I die of crucifixion and be buried, you'll resurrect me a few days later like Lazarus," but instead, he said:

Luke 23:42


  42 Then he said, “Jesus, remember me when you come into your kingdom.”


Question re-iterated: why is that in the Bible there's numerous records of people asking Jesus for earthly deliverance (thus believing that he had great power) &mdash; yet no record of asking him for eternal life?

Did their belief in Jesus's ability to heal them count as faith towards believing in Jesus and receiving eternal life?
You are right to point this out (sort of). If you read Mark's Gospel closely, you'll notice that the question of "who" Jesus is is of central concern. You'll further notice that whenever someone does get healed, Jesus will often tell them not to tell anyone. The only characters who know who Jesus is, are the demons that he exorcises, and, the centurion (aka John Wayne), and as you observed, one of the theieves. Scholars refer to this as the "Messianic Secret" For example:


  and he shouted at the top of his voice, ‘What have you to do with me, Jesus, Son of the Most High God? – Mark 5:7


In the above, oddly enough, Jesus does tell the demoniac to tell his friends what happened--but that is a matter for another question


  After sternly warning him he sent him away at once, saying to him, ‘See that you say nothing to anyone; but go, show yourself to the priest, and offer for your cleansing what Moses commanded, as a testimony to them.’ – Mark 1:43–44


Mark's version of Peter's Confession follows this trend:


  He asked them, ‘But who do you say that I am?’ Peter answered him, ‘You are the Messiah.’ And he sternly ordered them not to tell anyone about him. – Mark 8:29–30


Now, even these passages don't really say that Jesus was "God"—there existed many miracle workers in ancient times, and the claim to be Messiah was only to claim "anointed" status, like that of a king or priest.

Finally, in Greek, the word often used of Jesus' healings is sōzō, which actually has the connotation of "to save from harm." The idea is that Jesus not only "heals" physical ailments (perhaps using the word theropeuō), but heals the whole person—body and soul. Note in the Mark 1 passage that Jesus not only "fixes" a physical problem, but makes the leper ceremonially "clean," hence allowing him to worship in the temple. This is a common theme in Luke's Gospel too.

So, the long-and-short of it is this: did they "know" that Jesus was God? I don't think so. But that doesn't mean that they were "only" physically healed or that being physically healed is entirely separate from spiritually saved. Mainline Christians have been saying this for decades; it is the basis for the so called "Social Gospel."

Don't over-spiritualize this stuff, we aren't gnostics.