開戦といきますか！
Judging from the exclamation mark and from the context (a battle immediately starts after the commander of an army says this), I think that in the sentence above か could have an exhortative value. Is it correct? If so, is it a common thing or only a manga/anime thing? Thank you for your help!
I am going to say yes here even though "exhortative" may slightly be too strong a word for it.
The sentence-final 「か」 can certainly express inducement, solicitation, invitation, etc. and it is widely used in real life, not just in manga/anime.
The exclamation mark, however, is completely optional. It is just used more often in manga than in real life for the majority of native speakers.
"Let's start a war!"
"Let's go to a war!"
The final 「か」 adds a "shall we?" kind of nuance to the statement.
