Can I build on a different tile to the one I terraform?
No, you can only build on the space that you terraform. From the rules:


  First, you may change the type of one Terrain space. Then, if you have changed its type to your Home terrain, you may immediately build a Dwelling on that space.


If you are using the 2 free shovels power action, then you can use the second one to terraform a second adjacent space if the first one was enough to make your first space into your home terrain. 


  (If these Spades do not suffice to transform a given Terrain space into your Home terrain, you may exchange Workers* for the missing Spade – at the current Exchange rate on your Exchange track. If you only need one Spade to transform a Terrain space into your Home terrain, you may spend the second Spade on another Terrain space. However, you may not place a Dwelling on this other space.)