can you also tappa vikt or some such? or is gå ner i the best way?
Tappa vikt can be used.
Yes, "tappa vikt" can also be used but it may imply losing weight involontarily. For example, if you are sick.
English is not my main language but isn't "She went down in weight" also a good translation?
No, we wouldn't say this in English. We would say "She lost weight". Although my Swedish boyfriend has said "You went down in weight" and I obviously knew exactly what he meant.
Is this where "viktig" comes from? If something is important it has "weight"?
I thought the same...
Seems reasonable, since in English you can have "a weighty problem".