Gimpo is not "the domestic airport" as they have flights to Japan, China, and Taiwan daily from an international terminal. Very much so a real international airport haha. Much better to access and use IMO.
For WiFi Router - use KT
HOME | KOREA TELECOM . They have excellent English service and support in case of any issues (which shouldn't happen). You can prepay, reserve at nice discount and pick up easily at airport.
You absolutely do not need to overpay that HUGE price for car pick up. Even if you took a taxi from the taxi line at the airport, you would pay 60-70$ max for the taxi.
The bus itself is very comfortable, more so than a taxi IMO - bags go under the bus by the staff, English announcements, hard to make a mistake.
You can exchange your yen in Korea no problem. You can use your credit card essentially everywhere here EXCEPT to buy and charge your T Money transit card, at local markets, and at the tiny street newsstands. Taxis take them, can buy bus tickets, all restaurants, etc.