Now, I know the first thing that comes to mind is an assortment of tunnels and rooms. But I mean an actual city/town built fully underground, proper houses and such.
All with a entering tunnel and exit, basically, in summary, if you mine out a very large cavern is it possible to actually construct buildings without problems?
(I know I can test this but it takes a LOT of time to mine out a cavern big enough)
yes it is possible to do.
Perhaps looking in this topic Seraphia mountain town+rollercoaster design+sneak peak next project As you can see i,ve done it inside a giant mountain, but the same principle would apply to building underground.
Other things to consider are
-the know issue with water, stay away from any mayor water changing to prevent errors
-Farming can only be done on grass or dirt layer so you wont have that type of food below grounds
-mining out a mayor area takes a huge amount of time and can be heavy on your system.
-if there is space enemy’s will spawn below ground in normal and hard mode.
not sure about moving bushes or planting trees, but i don’t expect a problem there but that would be wise to check out first. It should not be to much work figuring that out 
frequent saving in different saves is advised
I have tried this before in previous alpha’s and it 's possible, but I always run into this one.
When I have mined to much…