Yes, it is 100% business use.
As far as I know, you are still allowed to use the Simplified Method for the Home Office, even if it is a separate building that is part of your 'home' property. That means yes, you have the OPTION to use the $5 per square foot method.
That method has the electricity, internet and depreciation of the building built-in to it, so you would NOT take the $5 per square foot AND the other expenses.
If you choose to use Actual Expense (not the Simplified Method of $5 per square foot), yes, you would prorate the electricity and internet. The method you described is the most common and is usually logical. For the building, you would depreciate the cost (your remodel expenses, plus any cost of the original building, such as the foundation, framing and roof).
Be aware that whenever you eventually sell your home, that separate business structure may not qualify for the $250,000/$500,000 tax-free exclusion for selling your principal residence.
