When Odo has another attack in the cargo bay and reverts to his liquid state, his combadge ripples and becomes a liquid along with the rest of his body. It may seem like the combadge is something that Odo might don after becoming humanoid, but there are numerous instances throughout the series that show Odo changing shape along with his combadge. The only conclusion is that the combadge is actually a part of his body and that he is able to imitate the electronics so it can work properly.
As Worf and Garak are fighting in the Jeffries tubes, a door to a Jeffries tube is knocked open and an unused area of the set can be seen behind it.
The founders were obviously very powerful, being able to assume any form. But they were organic beings and it wasn't made clear how they had the ability to turn Odo into a human.
It was a major error in judgment for the Founder to beam aboard the Defiant. If Sisko wanted to start a war, he could have killed or imprisoned her Jem'Hadar guards, and held her as a hostage. As the Jem'Hadar values her safety above anything else, they would not have attacked the Defiant, possibly giving Sisko a tactical advantage in destroying their home world.