first off, why do i need to keep it airtight
and finally, in my custom box, how would i install a terminal and still keep it airtight? some sort of sealant or what?

you need to keep it air tight, i'm not sure of the exact reason but i know you need to keep it air tight unless you are building a ported box. when you put a terminal cup on it you just need to seal it with silicone or something to that effect. you could also just drill a hole the size of the wire in the box and then seal it up with the silicone.

For the terminals, just use a hole saw, and either caulk, or silicone sealant, you get the idea. You have to keep it airtight because the air will change the way the speaker performs, just like ports change the way it performs. If you feel the need to, run some sort of sealant around the corners on the inside (if you can).

subwoofers are built (the cone) to produce bass. ok that was obvious but more technicaly it is litterally thrown up and down violently enough that if run w/ to much restriction or, on the other end of the spectrum, in an ambient pressure, it will not last you as long as you would hope! even a ported box provides restriction but obviously less than that of a sealed enclosure.
check the factory recommendations