You've got a lot of questions here, but let's start with the easy ones:
How far down would humans have to drill?
A Rapture Reminder
Depending on how they wanted to live, they wouldn't have to drill at all. Once the surface of the ocean froze over, the sea bottom would be insulated (but still freezing cold) and stay liquid for thousands of years, let alone a paltry two hundred. Deep-sea habitats, therefore, could be built and sustained indefinitely with geothermal or nuclear power sources. Farming would be strictly greenhouse-based, but you'd have plenty of energy to work with and warm them. Alternately...
Talkin' 'bout My Generation (Ship)
Again, no drilling necessary, save to mine the necessary materials - build arcologies or construct generation ships on the surface. It saves on geological engineering, and makes construction a heck of a lot faster. If you can build a self-sufficient space ship, you can build a structure on the surface to last two hundred years.
If you really want to use those neato (and terrifyingly fast!) laser drills, creating your own geothermal heat pipes would be a snap at half a mile a second.
In either case, light would be electrically generated, and water would be easy to find. (In the first case, not finding water would be the challenge!)
Preserving biodiversity would likely be a combination of advances in biology (allowing for in-vitro gestation, not just fertilization) and genetic engineering (to reintroduce heterozygosity when a lot of life will just have died).
Expansion is similarly easy - build more structures.
If you're really desperate to put people underground, at about a kilometre down things are a balmy 25 degrees, and that's not going to change just 'cause the sun switched off. That'll work just fine, and it'll only take you a second and a quarter to get there with those laser cannon... I mean drills.
As was pointed out in the comments to the main question, 200 years is a really, really short time for Earth to reach another star system, let alone slotting perfectly into the Goldilocks zone, so I'm assuming the same interstellar Thieves' Guild that stole our sun drops by two centuries later to try and undo the damage by teleporting Earth to an appropriate address.