Both ways work but it seems to be more idiomatic to say I don't understand anything in English and Jag förstår ingenting in Swedish.
No, that would mean like I don't understand whatever.
'not (just) anything' can mean vad som helst in some contexts, but not here. For instance 'I do not eat just anything' can be translated as Jag äter inte vad som helst. This kind of 'anything' means 'anything one could think of' but it cannot be replaced with 'nothing' – the sentence 'I do not eat just anything' does not mean the same as 'I eat nothing'.
Can this sentence also be used for not hearing something (e.g. in a loud environment)? In dutch you could use 'versta' only in that way (ik versta niks). Or is 'versta' a false friend?