NEX Tokyo Round Trip Ticket

Website said I can travel from Narita to Tokyo, and after getting off the Narita Express, SAME ticket can be used to transfer to other JR stations (am going to Shinjuku hotel)-- Is this correct?

Then can I use this same ticket (on same day) to travel from Shinjuku (after check into hotel) to other JR stations for sightseeing? Or do I need to purchase another JR pass for this same day?

The ticket is only good for transport as long as you don't leave the ticket gates, so you can transfer onto other JR lines to your final destination. If you leave the ticket gates, you'll have to pay for the rest of the ride. So to answer your question, no, it won't be good for "another" trip, especially after checking in to your hotel.

Web said in red
https://www.jreast.co.jp/e/pass/nex_round.html
After boarding the N'EX, passengers may transfer to other JR East trains to any JR East station within the designated Tokyo train area.
After exiting a ticket gate, the ticket cannot be used for re-entry.

That means oneway ticket valid for one day only NEX and after if want as no ticket gate out.
So NOT unlimted pass for all go/return day also for 14 days all.
If re-entry same or after date needs regular fare or paying by other pass(or IC card Suica/PASMO).
N'EX TOKYO Round Trip Ticket (1st one) validity already overed just very simple rule.
https://www.jreast.co.jp/e/pass/index.html

(note)
In Tokyo not only "JR" lines.
[PDF]
https://www.jreast.co.jp/e/routemaps/pdf/RouteMap_majorrailsub.pdf
white/black lines are all regular JR rail. (white/green : Shinkansen rail of JR-East)
other colored lines are all non-JR rail Subway,Private and more rail.
Open other rail and bus companies web discount ticket/pass pages.
http://en.wikipedia.org/wiki/Pasmo
Toei and Tokyo Metro are Subway(mainly underground) rail campany in Tokyo.
Only in Yokohama(not Tokyo/not connect from/to Shinjuku) city subway rail exist.

Some NEX trains go all the way to SHINJUKU.

a lot the nex trains go to shinjuku, so you don't need to transfer anywhere. Just ask for a ticket to shinjuku and they will give you the correct ticket.

Once you exit the ticket gate the ticket will be kept by the gate.

Does anyone know if the arrival and departure station can differ for the round trip ticket?

IE.
Arrival: Narita ー＞ Tokyo
Departure: Shinjuku ー＞ Narita?

Yes, they can.

When they mean "Tokyo" on the return portion to the airport, it really means any station with the N'Ex area.

The validity area is given here: http://www.jreast.co.jp/e/pass/nex_round.html#category05

Remember that connecting ordinary trains are also covered, as long as you don't exit the ticket gate (and remain within the validity area).