Yes, it's written "tallerkenen".
When pronounced, the second "e" is usually omitted, so it sounds like "tallærk'nen". In hurried speech, it can indeed be pronounced more like the ending of "mannen" as well, i.e. "tallærk'n'n".
Some may skip the first "n" as well (if they're accustomed to saying "en tallerk" in the indefinite singular), but this should be avoided as it makes you liable to repeat that pattern in writing, which would be incorrect Bokmål.
Yes, they're different letters in Norwegian, and never interchangeable.
If you're unable to install a Norwegian keyboard, you should still find clickable Norwegian characters below the input field in the browser version of Duolingo.
"Bok" is an irregular noun in Norwegian, which gets a vowel shift from "o" to "ø" in its plural form:
en/ei bok = a book
boken/boka = the book
bøker = books
bøkene = the books
Old English had the same irregular vowel shift that Linn mentioned. It used to be Bóc (a book), Béc (books). It's much like other irregular nouns such as Foot > Feet (Old English Fót > Fét). Norwegian, like English, keeps regular grammar so even with the vowel change it ALSO uses the -er ending. This would be the equivalent of English using Foot > Feets. I hope that explains it well enough.