I'd set up a fill clock and constantly add a certain score to a dummy objective. One score would then equal one tick. Using basic division, and some googling, you could get this done. Ticks in a minecraft day is very important in your endeavor. I would then set up a command block that removes x amount of points (ticks) from the scoreboard. I would also set up a command block on the SAME fill clock (all of this is on the same fill clock) that sets a certain block to a redstone block. This redstone block can be used to keep the door open for a few seconds, a few minutes, etc. The command block at the end of the elongated pulse, long repeater line etc. will fill the passage's entrance back up with a block.

Contact me if you need help that I did not give you here.

if you have 8 triggers you could set the 8th to put the value back to 0

To do it without command blocks, you could connect a daylight sensor to a pulse divider.

you can do that, or set up a cycle of, well, 7 days, and cause a minecart to advance to the next trigger each time it becomes night/day. That way you can easily control the order in which it happens.

If you're comfortable with commandblocks, do the same thing with a dummy scoreboard. Set up a commandblock to test each value from 1 to 7, and reset it as soon as it reaches 8. Then you can use the system suggested by Skitto to count it, but it'll reset to 0.

There's not really a way to do this in an easy-ish way.

You can set up a daylight timer, and whenever it goes off, it applies 1 of a score to a dummy player. When that dummy player reaches a certain number, you can apply the effects.

Use /scoreboard players test

I don't know about certain days but you could set up a timer that when it ends it runs a command or two