Do FS and Second Sunday are considered for "Mission Day Medal"? If no, What exactly are missions that count towards this medal?
Mission Days are special events that are separate from First Saturday and Second Sunday. During a Mission Day you need to complete 6 special Mission Day missions in a city that hosts the event on that day. You can check the locations and dates of the Mission Days until the end of the year here:
Once you have completed 6 of these special Mission Day missions you'll receive one point towards your Mission Day medal. You will only get the point if you do the missions on the day of the event however, if you do the missions after it, it won't count towards the medal.
There was an exception to this as when the Second Sunday was first introduced, completing six missions did give a point towards the Mission Day medal. I don't remember exactly how long that went on for, but eventually, Niantic introduced a separate Second Sunday medal after which it didn't count towards the Mission Day medal anymore.