Multi-language devving

Suppose i want to maintain French, German, English and Korean versions of my map at the same time.

What's the best way to do it, assuming i have important texts both in object and trigger data?

Simply translate the war3map.wts file.
If you're using JASS make sure you're using TRIGSTR references for strings that need translation.

I like that idea. I can export just files containing text and send them to translators.
However, the crux of my translation efforts will be unit/ability names and tooltips. Are these also stored in wts file?

Yes, they are there, that's the purpose of the .wts file.
GUI triggers and the object editor already externalize strings to that file automatically.
You only have to take care yourself when you're coding JASS.