"I think the dog has my baby."
Translation:Jeg tror hunden har babyen min.
Not a Norwegian native speaker here, but here's how I understand it (anybody, feel free to correct me if I'm wrong): You can use either mitt spedbarn or spedbarnet mitt (pay attention to the indefinite v. definite form of the noun depending on the location of the possessive pronoun). The only difference between the two is that mitt spedbarn somewhat emphasizes that it's my baby and not anybody else's while the other phrase lacks any emphasis and is neutral.
mener = think as in opine; be almost sure: "I think it's a good idea." "I think it was yesterday."
synes = think/feel as in opine; seem: "I think it's a good film." "It seems harmless."
tenker = think as in is engaged in the act of thinking: "I think, therefore I am."
tror = believe (both meanings), think: "I believe in God." "I believe that's the case." "I think so."
As you can see, there's quite some overlap. The most common mistake among English speakers, is using "tenker" when sharing opinions and doubts.