Bathroom fan install with new construction – lip around edge of fan doesn’t jive with ‘install drywall”

The three pairs of wires correspond to the three functions. You should be able to find labels or a note in the manual explaining which is which, but for now I'll assume:

• black/white: main light
• red/white: fan
• blue/white: night light

The green wire is a ground wire.

How you wire this up depends on what you have available. A 3-way switch will not help you here; you need multiple switches, and for that to work you may need more wires between your switch box and your fan/light fixture. Most likely, you will have a single cable (with a black, white, and bare ground wire) running to your fan. In that case, you can wire it up as follows:

• bond the bare ground wire with fixture green ground wire
• bond the supply white (neutral) with all fixture whites
• bond the supply black (hot) with the fixture black, red, and blue

That assembly will work, but it's not ideal as your fan, light, and night light will all turn on/off from a single switch.

A better version would be to have two switched wires running up to the fan. You can do this either by adding another 2-conductor cable (such as 14/2WG) running up to the fixture, or you can use a 3-conductor cable (e.g. 14/3WG) where there is a single neutral (white) and two switched hots (black and red). Assuming you have the 3-wire cable, your setup would be:

• bond the bare ground wire with fixture green ground wire
• bond the supply white (neutral) with all fixture whites
• bond the supply black (hot 1) with the fixture black and blue (lights)
• bond the supply red (hot 2) with the fixture red (fan)

This allows you to control the fan and light separately with your two switches, but again has the shortcoming that the nightlight is only powered when the light switch is on. Which probably defeats the purpose!

And that leaves the final, best option: two switched hots as above, plus an unswitched hot power supply wire (let's call it black-unswitched). In this setup, you'll

• bond the bare ground wire with fixture green ground wire
• bond the supply white (neutral) with all fixture whites
• bond the black-unswitched supply with the blue (nightlight)
• bond the supply black (hot 1) with the fixture black (main light)
• bond the supply red (hot 2) with the fixture red (fan)

This setup is ideal because you have switches to control your fan and light separately, and you have constant power to supply the night light. That means your fixture can use the unswitched power to run the nightlight constantly, and when you come in and turn on the main light, the nightlight either turns off or just becomes irrelevant.

As you can see, you have a few reasonable options for how to wire this fixture. Another variation is that you might have unswitched power in the fixture area, and only one cable running to your switch box. If that is the case, you'll be using the cable to the switch to both send power to the switch and receive power back when the switch is on. If you have that sort of set up, you could provide constant power to the nightlight even if you have a single switch controlling the fan and light together. You'll need to make a couple more junctions in/near the fixture in this case, involving the switch.

Note: a 3-way switch will not be helpful. A 3-way switch has three wires (call them A,B,C) and allows you to switch between connecting A+B or A+C. Two 3-way switches can be used together to allow switching a light from two locations. But as long as you have a switch with only two positions, you can only choose between on and off, not off, on-fan, on-light, and on-fan+light. For that you need two switches, or some sort of switch designed to control two separate fixtures.

Final cautionary note: To do this right, you will need some basic electrical knowledge, including how to properly connect wires with wire nuts and how to work on a fixture safely. If you aren't familiar with this sort of electrical work, you should get someone who is (experienced DIYer or pro) to work with you. Improper electrical work has a massive risk of hurting you, either while you're trying to install it or later on when a failure can put you at risk of fire. Good luck and be safe!

Probably the best solution in your case it to directly vent out. Since you have two external walls one of the walls will allow you to vent out of it. You need to pick the one that runs perpendicular to your joists.

There is not much downside in venting out. It is easier. By venting up you are actually causing an opening in your house for hot air to escape - hot air rises, will push open vent cover, and release. With a side vent you don't have those issues. The only issue is directing the moisture away from your house but this depends on the type of exterior you have.

You will pick a place for your fan - by the shower, in-between two joists. You will point fan vent towards wall that you can vent out of. You will run duct from fan to wall. Cut hole in wall slight bigger than duct (usually 4-6" in the US). You will cap it from the outside with a vent and secure vent to outside wall. Spray foam the openings/cracks created. That part of your fan install is done. Still need to get electric up to it.

To cover your other points. You can vent out the basement but you should not have a fan anywhere other than a ceiling. Anything higher than your fan will get condensation and could eventually mold.

Edit: Forgot to mention. Give your duct work a tiny tiny slope down so any moisture will drip out instead of back to your bathroom.