This is the working solution that I got (pictured below). I'll post my reasoning for each color beneath the image. Mild SPOILERS: It gives you a different ending, but It doesn't open the door, unfortunately.
SPOILERS FOR THE PUZZLE SOLUTION:
Going counterclockwise from 'F':
- F
- Orange: I got the solution from the "_15_" puzzle. If you count the items on each side (6 bottles on the left and 7 cups on the right) and then translate each number to letters (taking 15 as one number), you get 6=F, 15=O, and 7=G, making the word "FOG." "FOG" translates to the 'F' and 'G' in the puzzle, making the middle 'O' for "orange." That means the color for that "Lane" between 'F' and 'G' will be orange.
- G, Red, O, White: This is the next 4 characters. The map shows 4 nodes colored green, red, orange, and white, connected by a line. This can translate to "G - R - O - W." We have a 'G' and an 'O' already, making the two corresponding "Lanes" 'R' and 'W,' or "red" and "white" (I originally interpreted the orange on the map as yellow and the "white" color in the puzzle as beige, so I was a little unsure about this one).
- B
- Yellow: I used the pictures in the room to solve this puzzle. One seemed to be a picture of a rug, for "RUG," and the other seemed like a picture of rugby gear, for "RUGBY." I'm not sure I quite understand this one, but my guess was to take the word "RUGBY" and remove the word "RUG" from it, to get the word "BY." Using the earlier methods, we can translate the 'Y' to "yellow" and put it next to the 'B' in the puzzle.
- A
- Purple: For this color, I used the dessert menu. By taking the first, highlighted letters from "Apple Strudel," Pecan Pie," and "Eclair," we get "APE," meaning that 'P' goes in between 'A' and 'E' in the puzzle. Like previously, we can substitute "purple" for 'P,' giving us the last color, purple, in between 'A' and 'E.'
- E
I'm not sure if the door openable, but I couldn't find out how to open it. Maybe there's some other solution to the puzzle that opens the door, or maybe it's just a trick or decoration. Maybe there's something else to interact with that can open it? I have no idea, and I'm not sure how hard I'll try to find out.