To advance the day you could use the code I gave you originally and add the $gameDay:
<<if $gameTime == "Day">>
<<set $gameTime = "Night">>
<<else>>
<<set $gameTime = "Day">>
<<set $gameDate.setDate($gameDate.getDate() + 1)>>
<<set $gameDay += 1>>
<</if>>
Also this part of your code:
<<set $gameDay = setup.DAYS[] += 1>>
isn't valid code. You can't add 1 to an array with a missing index, which is what that code tries to do. I assume you meant to do this:
<<set $gameDay += 1>>
You could then display the day like this:
<<-setup.DAYS[$gameDay - 1]>>
Though a shorter way of combining those two lines would be this:
<<-setup.DAYS[$gameDay++]>>
because that would print the day based on the current value of $gameDay, and after that the "++" at the end would increment the value of $gameDay by 1 (a "post-increment"; as opposed to a "pre-increment" which, by putting the "++" in front of the variable, would increment the variable first, instead of last).
As for your <<link>> macro code, you're not quite using it correctly. I think what you're trying to do is this:
<<link [[Wait|living room]]>>
<<DaytoNight>>
<</link>>
That would show a link which says "Wait", and clicking it would first trigger your <<DaytoNight>> widget, and then it would take you to the "living room" passage.
Hopefully that answers your questions.
If you have any other questions, it's probably best to start a new question, since I don't always check back to see if there are new responses.
Have fun!