You are asking some questions related to the specifics of some given situations, where I am responding to the general question:
"Should tax dollars be used to finance Municipal Stadiums (baseball or football)?"
My answer to that question is: Yes. It can be a reasonable use of tax dollars to construct such a facility.
As to the specifics of who pays for what and suburbs vs. city or county, well, as the saying goes: "The devil is in the details."
But, I would say if a benefit can be determined, for a certain city, by the construction of a near by stadium in differeing city, then yes, there should be shared costs, profits and decisions.
Case in point: The LA Memorial Coliseum Commission. This entity runs the Coliseum and Sports Arena, which over the years has been home to the: Rams, Dodgers, UCLA, USC, Lakers, Kings and Clippers. It is run jointly by a committee of people appointed by the state, county and city.
They have also proved to be about the worst run agency of it's type in history. They have succeeded in driving off each of their major tenants, except for USC. However, the fact that it is a poorly run entity does not take away from the fact that there can be a system for shared revenues and costs.
MLB is a huge industry full of zillionaires. There is no reason what so ever for them to not pay for their own G*dd*mn stadiums. Not one red cent of taxes should end up in their coffers.
Same goes for farming, auto manufacturing, the airlines, trains, and energy production. Let the free market handle it instead.