As long as you plan
(not to go back to Tokyo but) to fly out from Kansai Airport [KIX],
a 7-day Japan Rail Pass would be too expensive
for your mentioned trips
(from Osaka to Kyoto, Kobe and Himeji and from Tokyo to Nikko).
on the Saturday (29th) I need to travel from Tokyo (Shinjuku) to Osaka. Where exactly is your starting point?
(What station on which line is near your Shinjuku accommodation?)
[Terms]
"Dryer" | "Drier" by Japanese hotels may well refer to a hair dryer,
which is never designed to dry clothes.
I suppose clothing dryers are not common in business hotels.
Some hotels may prepare a trouser press at guest rooms.
"Metro" in Japan refers to subway.
For this meaning, "subway" is commonly used.
Tokyo Metro is one of the two operators of Tokyo Subway Network,
which partly extends to the neighbouring Chiba and Saitama Prefectures.
[Tokaido Shinkansen]
According to Google Maps, travelling via Tokyo or Shinagawa is comparable in terms of time. Does it really make a significant difference? You would not have to make a detour via Tokyo Station, for any reason.
You highly likely can have a Shinkansen reserved seat
on Saturday, June the 29th, from Tokyo | Shinagawa to Shin-Osaka.
So, practically you would not have the merit of securing a non-reserved seat at Tokyo Station several minutes before the departure of your train.
Shinagawa Station has only one Shinkansen zone;
you just follow signs for the Shinkansen.
Tokyo Station has two separate Shinkansen zones,
with some gates of same or similar names.
You carefully choose a Shinkansen gate for the Tokaido Shinkansen,
which is indicated by a blue-and-white pictogram of Shinkansen train's head.
https://www.japan-guide.com/g3/2018_how_sign.jpghttps://global.jr-central.co.jp/en/info/station/_img/tokyo.jpgFor your information
: Early birds can take the only Shinkansen train starting from Shinagawa Station,
which is Nozomi 99 (bound for Hakata) departing at 06:00.
You can pay the JR basic fare
for the whole route from Tokyo (Ward City Zone) through to Himeji.
This basic-fare ticket is good for 5 consecutive calendar days.
At your transfer to the Tokaido Shinkansen,
you find a Shinkansen Transfer Gate.
This basic-fare ticket
does not let you make a stopover at Tokyo or Shinagawa,
so you cannot pass through a JR exit gate there.
[Luggage and Kansai accommodation]
should I just use a luggage transfer service? Due to G20 Osaka Summit, for a few days until June the 29th,
luggage delivery services for Osaka City may be delayed
and coin lockers at major railway stations in and around Osaka
may be closed.
One possible solution
is to stay in Himeji or in Kobe for your first Kansai night.
(G20 Summit is to be over by that night.)
For luggage delivery services,
usually there are limitations on the measurement and the weight of your parcel.
You had better send off your parcels
by the evening of Thursday, June the 27th.
With prior consent,
your hotel may keep your parcels until you appear at the reception.
[Himeji and Kobe combined trip]
In a case where you find a nice hotel in Himeji or in Kobe to stay for one night,
you can make a side trip in Kobe on your way from Himeji to Osaka or Kyoto.
For most tourist spots in Kobe,
Sannomiya Station on the JR conventional (= non-Shinkansen) line
would be more convenient than Shin-Kobe Station on the Shinkansen.
In Sannomiya,
there are also Kobe Sannomiya Stations of Hankyu and of Hanshin.
Among the destinations to which Shin-Kobe is a little closer
are Arima Onsen and Kobe Nunobiki Herb Garden.
For Himeji,
you might hope to go by the San(')yo Shinkansen from Shin-Osaka,
especially during a weekday morning commuters' rush.
Track 20 of Shin-Osaka Station,
which may be used by Shinkansen Mizuho, Sakura and Kodama trains,
can be a little tricky.
The Shinkansen concourse on the 3rd floor
and the platform for Track 20 on the 4th floor
are connected by two elevators;
you change elevators on the mezzanine.