Is the duct sheet metal or flex duct? I've never seen prefabricated dryer ducting that wasn't 4". How does it mate to the dryer, where the outlet is certainly 4"?

If they're really different diameters you should be able to join them with a good quality, metal backed duct tape. Or if it's flex duct, replace it.

I believe it is flex duct. It looks something like this

Here is a link that might be useful: flex duct

That should fit - notice how the ends are slightly smaller than the middle.

Yes, I would purchase and install an auxiliary lint box*. Also, are you having this installed by a licensed electrician?

*Link provided solely as a reference and not a recommendation. I have no affiliation with said company.

But my dryer already has a lint filter. Do I still need a auxiliary lint box?

I've not yet seen a dryer filter screen that blocks all the lint; this can be substantiated by the amount of blow-by lint found in the dryer ducting when you go to clean or replace the ducting on even the shortest of runs. The length of duct from the dryer to the auxiliary lint trap will become the most lint laden of the ducting, but it is easily cleaned; the balance of the thirty feet much harder to clean. The associated dampness of the dryer's exhaust is the primary catalyst to the lint formation on the inside of the ducting. You're running thirty (30) feet of ducting and now it is further assisted by a booster fan at the terminating end. Without the auxiliary filter you will be pulling more lint through the duct increasing the chance of lint formation.

I installed a lint buster fan. No additional filter needed.

Here is a link that might be useful: Lint buster fan booster