In this thread, I'll be counting room transitions starting from and including the click on the door out of the entrance room, up to but excluding the click home. This corresponds to all room transitions that are fully included under the current timing rules, given that the first frame of the click home is considered the stopping point.

I count 76 transitions in Kjorteo's current 6m56s900ms 100% (Hardware) run, using a route that starts in King's Court and zigzags north-south-north while going west-east, with a detour into the Elemental Zone. In my own current 4m21s467ms 100% (Emulator) run, I count 78 transitions (including some substantial navigational mistakes), using a route that starts in the Alley, then goes north along the Market corridor and south along King's Court before crossing back east. I did this without a map at the time, and there are some paths I thought were closed that seem to not be.

I believe a route with 72 transitions is possible starting from the Alley Entrance:

1. (1) N to Alley South.
2. (5) E SE N W W to clear Locust Loop.
3. (3) W W W to clear Parrot Way and reach Market South.
4. (2) S N to clear Market Entrance.
5. (8) NW N S W S W N W to clear the Produce Zone and the Elemental Zone and reach King's Court South.
6. (2) S N to clear King's Court Entrance.
7. (3) SW N E to clear Rug 'n' Mug and Passerines.
8. (3) N E NW to sweep up the rest of King's Court via Shawlbeard.
9. (3) W W W, entering the Color Maze.
10. (6) W W N NE E S to loop around the Color Maze.
11. (3) E E N to reach the Golden Hall.
12. (5) N E E E E past the Prim Redhead and entering the Blue Marble Zone from the west.
13. (4) N E S S to clear the rest of the Blue Marble Zone and re-enter the Market at Market North.
14. (8) S W E SE N N E E to clear out the rest of the Market and traverse the northeast passage to Alley North.
15. (3) S W E to grab Chained Redshirt and prepare for the final stretch.
16. (7) SE E N N S W N to swing through the Tunnels and reach the Shining Door.
17. (4) N E W S to clear out the northeastern L and return to the Shining Door.
18. (2) W N to reach the last lousy room and wave goodbye to Peeking Orange on the way home.

This takes the western half of the map in the opposite order from my last run and avoids some backtracking in the Market/Alley crossovers.

A route that visits all rooms requires at least 69 transitions.

We start by assuming that we have a full route, then count and remove groups of required transitions, along with removing rooms that are cleared (both entered and exited) by those groups and counting rooms that are guaranteed to be entered (but each one not more than once). Note that in many cases a group of rooms can be used either as a loop or as a thoroughfare; the minimum number of required transitions assuming the rest of the route is maximally convenient is what is listed.

We start with 58 rooms, which I'll use abbreviated names for many of because otherwise oh dear. {KC,M,A}{S,M,N} are the three major south-north paths in the King's Court, Market, and Alley. T{L,H,R,G} are the Ladder, Hermit, Rats, and Gate in the Tunnels. SD is the Shining Door. P{S,P} are Produce Storage and Produce Passage.

The groups are as follows:

• Entrances: You start at one entrance for free, but it requires a transition to get out of. The other two require two transitions each. +5 transitions, −3 rooms, guaranteed entry to KCS, MS, and AS.
• Color Maze Loop: From the Safe room, there is only one way to enter the Color Maze, and you must leave by the same path, taking four transitions. It takes six transitions to clear the Color Maze. +10 transitions, −7 rooms, guaranteed entry to Safe.
• Rug'n'Mug Loop: From KCS, you must clear the loop of two rooms to the west and return to KCS. +3 transitions, −2 rooms, guaranteed entry to KCS (redundant).
• Crone End: The only way to clear Brawl and Crone is a backtracking loop starting and ending at SD. +4 transitions, −2 rooms, guaranteed entry to SD.
• Red and Blue: The series of rooms from Grass Corner through Blue Marble Southeast can be treated as a straight path.¹ +8 transitions, −8 rooms. No guaranteed entry can be tacked onto the end because Grass Corner is a map exit.
• Golden Hall: Must be entered and exited. Safe and Grass Corner are already known to have been visited. +2 transitions, −1 room.
• Nooks: PS, Purple Rug Lady, Chained Redshirt, and TL are all side rooms. +8 transitions, −4 rooms, guaranteed entry to PP, MM, AM, and TH.
• Elemental Zone: Has KCS to the west and PP to the east, both known visited. Can be a loop or a thoroughfare but takes a minimum of five transitions either way. +5 transitions, −4 rooms.
• Parrot Way: Has MS to the west and AS to the east, both known visited. +3 transitions, −2 rooms.
• Locust Corner: A required loop, and AS is known visited. +5 transitions, −3 rooms.
• Shawlbeard Cluster: Shawlbeard, KCN, and KCM. Adjacent to Safe and/or KCS, both known visited. +4 transitions, −3 rooms.
• Tunnels South: TG and TR. Adjacent to AM and/or TH, both known visited. +3 transitions, −2 rooms.
• Tunnel Ladders: Must be entered and exited. Adjacent to AM, SD, TH, all known visited. +2 transitions, −1 room.

This is a total of at least 62 transitions to clear (enter and exit) 42 rooms. An additional 9 rooms (KCS, MS, AS, Safe, PP, MM, AM, TH, SD) are guaranteed to have been visited as a side effect. That leaves a cluster of 7 rooms (MN, Crumbling Corner, Teal Lady, Goat Field, Goat Corridor, AN, Peeking Orange) which is a bit harder to analyze due to the remaining two map exits being in there along with various harder-to-constrain links to their surrounding rooms. Nonetheless each of those 7 rooms will take at least one transition to enter, for a total of at least 69 transitions. This does not prove that a 69-transition route exists, only that a route with fewer than 69 transitions is guaranteed to miss at least one room.

¹ I conjecture that it is never transition-optimal to use the southern east-west door in the Blue Marble Zone, though I haven't constructed a proof; a route where that door weren't easily avoidable would involve using the Blue Marble Zone as a loop rather than a throughfare, and that results in uncomfortable backtracking through Prim Redhead. This may however be of relevance to non-room-complete paths, where maybe you hope that the middle three rooms contain no sheep…