The subway takes you very near. I can’t remember the name of the stop but the museum website should tell you.

They normally do. But, there are days they close.

You can take Line 4 (Jungang Line) to Ichon station. Get out at exit 2 and walk towards Yongsan family park. It is open daily from 10am to 6pm. It is open for longer on Wednesdays and Saturdays. On Sundays it is open up to 7pm. Your hotel is located between City Hall and Seoul Metro stations - and you can ask the concierge staff at your hotel which is the best station to use to get to Ichon station.

The museum fee is free for the permanent exhibition, and they only charge for special exhibits.

Вход в музей бесплатный. На специальные выставки - платный вход.

You should take line 4 to get to Ichon station. Then take exit 2 and walk for about 200 metres heading towards the Yongsan Family Park.

Yes, there are boxes to keep your belongings, however the luggages can leave at the counters. There are museum staffs helping you for this.

Not a cultural asset, you can take it out.

Most museums in Seoul are free (Modern Art, an exception). However, there are special exhibits running within that require a fee.