Is a Special Action required to conduct breakthrough combat and then another one for breakthrough movement? Or does 1 SA buy both?
You can spend 1 SA for a Breakthrough Operational Movement Phase and then have a mandatory round of combat (without spending any other SA) only if the area designed is now contested.
See rule 6.73: If a Breakthrough Operational Movement Phase ends with an area being newly contested, a mandatory combat round follows and must immediately be fought for that area. This mandatory combat round does not cost a Special Action, other than the one already spent for the mmediately preceding Breakthrough Movement Phase. Breakthrough Movement into already contested areas do NOT result in combat as part of the same Special Action.
Does Mud prevent Breakthrough Combat from occuring?
I assume no because Mud prevent only Breakthrough Movement Phases as per 1.13: … No Breakthrough Movement Phases (Operational or Strategic) are allowed into or out of any area experiencing mud. Armor may never use any hit-bonuses in mud (attacking or defending), including the surprise attack bonus (see 14.43(b)) and the elite unit bonus (see Glossary). Defending infantry-class units, receive a hit-bonus during mud turns. Ground-support units may not participate in combat in an area experiencing mud weather.
Can Out of Supply units conduct Breakthrough Combat with a SA?
I assume yes because OOS cannot have Breakthrough Movement Phase but roll half the total dice allowed when attacking
See rule 3.4: Units that are out of supply may move only one area during the Initial Operational Movement Phase; may not Strat-move; may not move at all using Breakthrough Movement Phases; may not receive replacements; and roll half the total dice allowed when attacking (rounded down).