I recall it is available but I don't know the cost. Better to email hotel directly to ask.

Just over 30km. Used to have an airport bus that ran directly from Incheon Airport to a drop off directly outside the hotel but the route hasn’t restarted since the pandemic. Easiest option is a taxi from Incheon.

2 options, you can ask at the bus kiosk if there is a bus servicing the hotel, I think it depends on hour and frequency (if you can wait). The other is to get a taxi at the middle island of the street going out of the terminal. There should be a taxi stand and people will guide you , depending on how far you need to go. Credit cards accepted most of the time.

The bus service to Incheon Airport is done with a KAL limousine bus which is stopping just outside the hotel. As it is a KAL service, it will stop at the "Korean Air" terminal (T2) first before coming back to T1. If you have to go to T1, using the KAL limousine bus therefore takes an unnecessary long time. Unless you have plenty of time I'd recommend to use a taxi. By the way, the subway is not really convenient from Suwon as it involves several transfers. I'm not sure, however, if there's a better option.