When are Mice and Mystics encounter cards flipped & discarded?
What if we have not actually had a surge and we go to a new tile, which has a special encounter, should the previous encounter card be discarded?
Yes, the previous encounter card is discarded. From the rule book

Encounters - Each time a mouse explores an area that has not already been explored during that chapter, the mice will have an encounter. Unless otherwise stated in the chapter being played, draw a card from the encounter deck. Each encounter card has multiple potential encounters based on how far the chapter has progressed. Use the encounter that matches the page space that the hourglass marker currently occupies on the story control board. Place the new minions using the placement rules below, then determine an initiative order. Once the minions are placed, place the encounter card face-up on top of the encounter deck. If there is still an encounter card face up from a previous encounter, discard the old card. This new face-up encounter card will now determine the next non-chapter-specific surge effect.

What if we have a surge on a tile that has a special surge, but we already have an encounter card present?
You perform the special encounter, but you don't discard the encounter card.

Room tile rules: some room tiles that are being used in a chapter will have specific rules that apply to that room. Each room tile that has special instructions [...] or special surges that instruct players on what minion(s) to place when a surge happens in that room (special surges take place instead of a surge listed on an encounter card). (page 5, Story Book)
[Step 3] If there are no chapter-specific surge instructions, follow the surge effect on the face-up encounter card that is on the encounter deck and then discard that card. (page 15, Surges)