You are correct about Swedish. But a non-Swedish text may contain Swedish parts, dialogue by a Swede for instance, and then you might find ä and ö.
Danish uses "at" where Norwegian (both forms) uses "å". It is the infinitive marker, used much as "to" in English. Danish isn't very fond of diphthongs either.
Nynorsk and Bokmål is harder. The vocabulary of bokmål and nynorsk are not identical. The personal pronouns for instance are not all the same, here's some of the differences:
1st sing 1st plural 2rd plural 3rd plural
Bokmål: jeg vi dere de
Nynorsk: eg vi/me/oss de/dykk dei
Some users of Bokmål collapse the feminine and masculine genders into a common gender, using the inflections of the masculine.
Nynorsk inflected words are less likely to end with -n, or -t and more likely to end with a vowel.
Nynorsk in general uses more diphthongs, and different diphthongs in some places. For instance in some strong verbs the past tense forms uses different diphthongs:
infinitive present simple past past perfect
Bokmål: å krype - kryper - krøyp/krøp - har krøpet
Nynorsk: å krype - kryp(er) - kraup - har krope/kropi
(The dialect I speak most of the time uses
å krype - kryper - krøyp - har krypi
Some might consider this to reveal my political stance :)
Good Nynorsk can be quite different from Bokmål grammatically but Nynorsk translated from Bokmål uses Bokmål grammar. I know good Nynorsk when I see it but I don't get to see it very often so I can't give you any "look for this" tips :)
Disclaimer: I am Norwegian and have not studied Norwegian after high school. I have read a reference grammar though, in order to write a recognizer: Faarlund, J.T. Norsk Referansegrammatikk. Universitetsforlaget, 1997.