Any character, including a henchman, is able to have structures built. Only PCs of name level can benefit from domain rules, however (e.g. tax, followers, monster-clearing, new settlements, etc.).
Costs of building contents and furnishings do not need to be painfully considered. The player should just propose something that seems reasonable and not spend hours wondering if they've properly priced their dining chairs. Review the prices of items in the PHB and any other sources, and make a fair guess.
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