Hi there.
About your additional question...
It seems you are planning to trail mt.fuji AND outlet AND heading back to Shinjuku on the same day.
...I'm really sorry but it's impossible.
If you want them all in the same day, you have to give up to trail...take a close view of the mountain from the Kawaguchiko station.
And if you want to trail, yes, as you've wrote, you can take the bus from Shinjuku station to the 5th station of mt.Fuji ( called Yoshida-guchi st.) 1 stop ahead from the Kawaguchiko station, operated by Keio Bus on every weekends and holidays.
Leaves 7:45 and 9:45, arrives 10:10: 12:10.
You can purchase at Shinjuku station anyway.
Of course you can purchase any tickets at Kawaguchiko station, or at the outlet too.
Kawaguchiko station is a terminal station around the mountain.
Gotenba outlet is HUGE, though it never be bigger than Fuji...;P
I recommend you to take at least 1 day to trail, 1 day to Gotenba if you stay Shinjuku at night.
There are direct limousine buses from Shinjuku to the outlet every day.
Costs you 1,650 yen for 1 way, 2,880 yen for round trip.
( I, to be more honest, recommend you to take 2days in Kawaguchiko area if you are going to trail...then Gotenba)
Hope this can be your help!