Your assumption seems to be right (although how widely it works, it's hard for me to say, maybe all such words, maybe there are exceptions), but for masculine and neuter nouns. kotem/lwem/psem/robotnikiem; drzewem/dzieckiem/krzesłem/jajkiem.

If it was your brother who is British, it would work: Mój brat jest Brytyjczykiem. But as the nationality is a noun, and it also takes gender into considerations, its Nominative is Brytyjka, and "Moja siostra jest Brytyjką " (/Angielką/Szkotką/Irlandką/Walijką/Francuzką/Włoszką/Węgierką etc.)

I read that "to be" takes Narzędnik, which is usually -em.

Awhile ago I bought this, which has some of the basics about the cases: https://www.amazon.com/Polish-Verbs-Essentials-Grammar-Oscar/dp/0071597468/ref=sr_1_2?ie=UTF8&qid=1472135950&sr=8-2&keywords=polish+mcgraw+hill

It is very basic but I guess it is nice to read about the cases.

I like to use this site when I am not sure about which case to use: http://aztekium.pl/przypadki.py?lang=en

Theoretically you can (added "Moja siostra to Brytyjka" now), but we don't really recommend this construction when you have a person on the left side of "to". It sounds a bit clumsy, more like "My sister = a British woman" than a descriptive statement that "My sister is British". But we do usually accept it. What we don't accept, by the way, are things like "Ona to Brytyjka". If you have a personal pronoun on the left, it just sounds too clumsy.

Tak, rzeczowniki określające narodowość (Polak, Niemiec, Brytyjka, Ukrainka) piszemy dużą literą.

Przymiotniki (polski, niemiecki, brytyjski, ukraiński) piszemy małą literą.

Yes, indeed. Using the adjective would be a mistake here.