This is mainly for the two puzzles.
For the color puzzle, you need to click in the center of each group of four. So think of the center of those nodes: 2 nodes in each of three rows, between the four rows of colors. Click like this:
3rd node row, 1st node, click twice.
2nd node row, 2nd node, click three times.
1st node row, 2nd node, click twice.
1st node row, 1st node click twice.
2nd node row, 2nd node, click once.
In the arcade puzzle, the way the doors work is tricky. This solution is a bit convoluted. There may be a simpler solution, but this works.
You can use your d-pad. You don't need the screen buttons. I'm typing this instead of using button icons for more clarity.
U4, R3, D2, L3, D2. You should be back at start with 3 open doors.
R3, U2, L3, D2. You should be back at start with another open door.
R3, U2, L3, R3. This closes the center door.
D2, L3, U4, D4. You should be back at start with everything the same except the top right door is now open.
R3, U4, L6, D4, R3. You should be back at start, but the bottom left door is now open.
U4, R3, D2, L3, R3, D2, L3. You should be back at start.
U4, D4. This closes the top door.
R3, U4, L6, D2, U2. The finish is now open. Now we just need to circle around.
R6, D4, L6, U2, L1. Voila!
Hope this helps with that part.