For public services and non profit, here you say you think wages are the only component included: "But in many cases those services represent the value. So what winds up being included is just the wage." When you say "just the wage" it means you don't think other costs are included (materials, overhead).
I speculated that the total cost must be used. Then you say the truth is exactly like you say but the quote you provide says this "the value of government output is deemed to be exactly equal to the total cost of the wages paid to public servants plus the cost of intermediate inputs."
If you look up the definition of intermediate inputs you'll see it includes other services, goods, and energy. So in the case of nursing they include medical supplies, electricity, and office costs like payroll services. It doesn't seem like that's what you said in your first post.
Regarding dividends, they are a distribution of profit. They are not a cost. Shareholders own a portion of the company. In a privately owned company the owner may transfer money from the company account to his personal account. This is not a cost for the business. In the same way, payments of dividends is a distribution of profit to shareholders.