I believe there is. I saw lots of families parking them as they got in line to ride the cable car to the top.

It depends on your pace. About 45 minutes-1 hour.

Entrance fee Namsan Park is free ( walk on feet) and Cable Train return ticket is 8500 won/ person

Yes

頂上の工事のことでしょうか？わかりません。

Yes it still have some, but most of them had fall already. The best day to see them is 5 Apr - 15 Apr

You don't take a taxi because the fee is expensive. You should take subway line 3 or 4 at Chungmuro station, go out of exit 2 to the right of Daehan Multiplex entrance - take the yellow bus 2 at the bus stop in front of the subway station - get off at the N Seoul tower bus stop