TL;DR: In principle, costs for a hotel are not part of the compensation and have to be reimbursed separately. However, according to EC 261, the airline does not have to pay for a hotel at the destination after arrival. You may be due reimbursement according to Norwegian law or the Montreal Convention, though.
Your question, in my eyes, is two-fold.
Firstly, you are wondering whether care according to Article 9 of EC 261/2004 can be considered part of the compensation according to Article 7. This is easy to answer: No, care has to be provided in addition to any due compensation. Indeed, Articles 7 and 9 make no mention of each other and can thus be assumed to be both applicable independently. This has already been established on Travel.SE (although that was about meal vouchers, not hotels) and the regulation is interpreted in the same way by experts on the matter:
When take-off is delayed by at least two hours, the airline is compelled to take care of you. Your "right to care" is an obligation for the airline to provide [...] accommodation and journey between the airport and the hotel, should you be delayed overnight. [...]
In addition, and more interestingly, if your plane arrives at your final destination 3 hours or more after what was initially planned, you can claim compensation.
In case the airline refuses to provide accommodation and a passenger has to pay for a hotel himself (it is unclear whether you asked Norwegian for a hotel after arriving in Rome), the Interpretative Guidelines on EC 261/2004 clarify that
If care is nevertheless not offered even though it should have been, passengers who have had to pay for meals and refreshments, hotel accommodation, transport between the airport and place of accommodation and/or telecommunication services can obtain reimbursement of the expenses incurred from the air carrier, provided they were necessary, reasonable and appropriate.
This reimbursement takes the place of the care the passenger should have been provided, and thus does not impede a possible right to compensation. Hence, Norwegian's claim that "this EUR 36 is included in the EUR 400" is false. (Otherwise, why would the money for food not be included in the EUR 400 as well?)
Secondly, however, one should question whether you even had a right to a hotel upon arrival. So let's have a closer look at Article 9:
Where reference is made to this Article, passengers shall be offered free of charge: [...] hotel accommodation in cases where a stay of one or more nights becomes necessary, or where a stay additional to that intended by the passenger becomes necessary;
Well, a stay of one night was indeed necessary in your case, but where can one find references to Article 9, as required? In your case, relevant references are found in Article 6, which lays out your rights in case of delays:
When an operating air carrier reasonably expects a flight to be delayed beyond its scheduled time of departure:
(a) for two hours or more in the case of flights of 1500 kilometres or less; or
(b) for three hours or more in the case of all intra-Community flights of more than 1500 kilometres and of all other flights between 1500 and 3500 kilometres; or
(c) for four hours or more in the case of all flights not falling under (a) or (b),
passengers shall be offered by the operating air carrier:
(i) the assistance specified in Article 9(1)(a) and 9(2); and
(ii) when the reasonably expected time of departure is at least the day after the time of departure previously announced, the assistance specified in Article 9(1)(b) and 9(1)(c);
So hotel accommodation according to Article 9(1)(b) only has to be offered if the flight departs on the following day, which didn't happen here. Again, experts come to the same conclusion:
If your flight is delayed or cancelled and you have to stay at the place of departure for one or more nights, you are entitled to accommodation and the airline is required to inform you of this right and to offer you a place to stay.
End of story? Not quite. There was a similar case in Germany, where a passenger travelled from Hamburg via Frankfurt to Bilbao on Lufthansa, missed his connecting flight in Frankfurt, and the next flight to Bilbao arrived so late that he had to stay in a hotel near Bilbao airport for one night. Lufthansa used the same reasoning that I have given and claimed (translation mine):
Für Ankunftsverspätungen sieht die VO (EG) Nr. 261/2004 gerade keine Versorgungsleistungen vor.
For delays on arrival, EC 261/2004 does not stipulate any right of care.
Nevertheless, the passenger sued Lufthansa for the cost of the hotel and two taxi rides, and won! The court argued as follows:
Diese Kosten sind jedenfalls gem. §§280 Abs. 1, Abs. 2, 286 Abs. 2 Nr. 1 BGB ersatzfähig, da sie adäquat kausal durch die Verspätung der ersten Beförderungsleistung verursacht wurden.
These expenses shall be reimbursed in accordance with §§280 (1), (2), 286 (2) No. 1 of the German civil code, since they were caused by the delay of the first transport service.
So the judgment was made based on German national law, but don't be disappointed:
Dahinstehen kann auch, wofür vieles spricht, ob es sich um eine internationale Beförderung i.S.d. Art. 1 des Montrealer Übereinkommens (MÜ) handelt und damit Artikel 19 MÜ Anwendung findet. Es würde sich hieraus nichts anderes ergeben.
It is irrelevant that this journey probably constitutes "international carriage" according to Article 1 of the Montreal Convention and therefore Article 19 MC applies. The result would be the same.
Indeed, Article 19 states
The carrier is liable for damage occasioned by delay in the carriage by air of passengers, baggage or cargo. Nevertheless, the carrier shall not be liable for damage
occasioned by delay if it proves that it and its servants and agents took all measures that could reasonably be required to avoid the damage or that it was impossible for it or them to take such measures.
and a passenger having to pay for a hotel upon arrival due to a late flight is, of course, damage.
Bottom line: You can't be reimbursed for the hotel stay according to EC 261, but a reimbursement is due according to the Montreal Convention.