Depends on the airplane and configuration.
The 737 is a fail-passive autoland system as it has only two autopilots. At 500' radar altitude, a nose-up trim input is applied, called the pre-flare, which puts the airplane is an out-of-trim condition until in the flare. At that point, you are not allowed to disengage the autopilot due to this out-of-trim condition and, if a TOGA button is pushed, the autoflight flies the go-around. If both autopilots are NOT engaged, you are not in the autoland configuration and no pre-flare trim is applied. In that case, the single autopilot is not able to perform the go-around so pressing a TOGA button will disengage the autopilot and the go-around is flown manually.
On the 757/767, and I assume the larger Boeings, you have three autopilots (left/center/right) and it does a fail-active autoland. There is no pre-flare trim. When the ILS is captures (I forget the details of the criteria), all three autopilots become active including full rudder control, in place of just the standard yaw damper. Pressing a GA paddle, at any time on the approach, will have the autopilot fly the missed approach.
They don't.
MDA's are converted into Derived Decision Altitudes (DDA) by adding 50' to the MDA. Published VNAV DAs are not changed.