As far as I can tell, each "mine hole" is unique. I went into a 2nd hole, to see if it would start me in a different spot in the cave, but it generated a new cave. I liked the 2nd cave better, so I never went back into the 1st one. I don't know if doing so would put me in the same cave I started to explore or if it would generate a new cave again.

Caves are definitely easier than Maxwel's Door / Adventure Mode. I have yet to complete a single chapter in adventure mode.