2 - Sisko chose the defiant in 'the search' - and then he could most definitely choose his ship.
Different circumstances. When Sisko asked for the Defiant
before "The Search," the ship was an unfinished and largely untested battle cruiser (recall that the project was abandoned by Starfleet before Sisko revived it). The vessel was something of a lemon and was not highly regarded by Starfleet. Sisko got the Defiant
not because he had the authority to choose his own ship, but because he had the virtue of wanting a ship that no one else did.
As you pointed out, the Defiant
performed well while stationed at DS9 both before and during the Dominion War. As such, other vessels of the same class were put into production because Starfleet recognized the value of having ships exclusively devoted to combat available for use. The Valiant
was used as a training ship for cadets; the Sao Paulo
was built, as well.
When the Defiant
was lost, Ross told Sisko he'd find another ship for him, but he didn't specify when or what type of ship it would be. That DS9 received another Defiant
-class vessel may have been the result of Sisko asking for it, though there's no evidence suggesting anything like this took place. Dax herself said that she didn't even realize another Defiant
-class vessel would be in the mix (though she was absent for much of the intervening time period). OTOH, it may just have been a huge coincidence. The war was still going on, so it's not as though Sisko had time to think it over once a new ship was available.