Does this mean train as in work out with weights/sports practice? Or could you say 'jag träna min svenska språksförmåga'?
It can mean either. If used without an object, it's commonly referring to the weights/sports thing.
What about "My wife hates doing exercise?". As a non-native English, I see it a as good alternative.
That could also be "My wife hates to do exercise", but has an extra verb in there, "do", and "exercise" is a noun in that sentence. I believe that, translated into swedish that would be "Min fru hatar att göra träning"
I put "My wife hates to DO excercise" and got it wrong because of the do
The variants "doing exercise" and "to do exercise" are both fine, and I'll go add them right away. They're not strictly great translations of the Swedish phrase - but we don't have a counterpart for that phrasing with a noun in Swedish, so we do need to include them. The suggested att göra träning does not work at all.