Eighteen Eighty-Five Solution
Author: Chris King
In order, the answers in this round are
As Doc notes, he feels that he "shares" a lot with the surroundings here in the West, and the map prominently features buildings.
With that, it is shown that each of these answers shares the same length with one of the buildings on the map, while also sharing two letters with those answers:
The first set of letters spell THIMBLE, and the second set create THEATRE, so this round's item is the THIMBLE THEATRE. TO discover which building on this map is the new HQ, connecting the dots results in...
...an arrow pointing right at the POULTRY SHOP. Typing all of this results in the capture of the Bioshock Infinite character Elizabeth, who is known for her time-traveling with tears in the continuum ability, her bird, which would be kept in the Poultry Shop, and her home town of Columbia (not SC, but the floating one in the sky).