As the title says, I'm wondering if a species could theoretically boost an asteroid near a planet for easier mining access. If this was to accidentally crash into the original planet could a habitable moon form, even over a long timeline?
Based on the knowledge we have, I would say the answer to your question is "more likely not than yes".
Explanation: our Moon is thought to have formed when a Mars-sized body (way bigger than an asteroid) impacted at low relative speed Earth, spraying around part of the crust and mantle, which later coalesced to create what we today know as the Moon.
In this process lightest elements were favored (as they were more concentrated in the outer layer of Earth), resulting in a less dense Moon.
If you couple lower density and lower size you automatically get lower surface gravity which, needless to say, make retaining an atmosphere really cumbersome. Also light elements do not really build an efficient core dynamo, so you can wave goodbye to a long lasting magnetic field protecting whatever flimsy atmosphere your newly formed body may have caught.
No atmosphere is surely not a good premise for life, therefore I hope your spatial mining engineers do their math well when they are boosting that asteroid for mining.
If the asteroid was big enough that the impact threw enough matter into orbit it could form an accretion disk. Given several billion years the accretion disk could theoretically become a moon. As for whether it was habitable or not I will be forced to assume the alien species you are describing holds the same definition of habitable that we do, and will define it as possessing a breathable atmosphere, plentiful water and flora and fauna. This would take several more billion years and require that the planetoid that formed was geologically active with a high iron content for a core so as to maintain a magnetic field capable of shielding the surface from cosmic radiation. If the planet maintained surface water it is theoretically possible that microscopic extremophiles such as prions could have survived the impact and provide the seeds for life to arise again. The moon would also need to remain large enough for its gravity to maintain its liquid water and atmosphere. Saturn's moon titan has a gravity a little lower than our moon and has managed to hold on to a very dense atmosphere, albeit one that is not breathable.
Take a look at this.
The premise: a massive asteroid comes in at a very oblique angle. It decelerates into the atmosphere and strikes the planet a glancing blow. The asteroid winds up in orbit, not much worse for the wear. The energy to decelerate it, though, cooks off the atmosphere in much the way the dinosaur-killer impact was thought to do.
Maybe that atmosphere gradually recoalesces - around planet and asteroid. I suspect the asteroid would collect some of this atmosphere too.
Another thought about the asteroid that becomes a moon - the asteroid would also heat up as it decelerated thru the atmosphere and skipped off the surface. You could have frozen materials in the interior. As the red hot exterior loses heat to the cold interior these materials would outgas. This phase change would absorb heat and let the asteroid cool quickly. Also these gases could then constitute the atmosphere of this new moon.