"She went to Beijing a year ago."
"她一年前去北京." is a valid translation for this. The additional words are not necessary.
Yes. The only scenario we would say it is when somebody ask precisely for the time she went there. The emphasis would then be on "a year ago".
This sentence seems incomplete to me. You are expected to say more if you say this. Or you can say "她一年前去了北京", then people will know that you have finished this sentence.