Do people use 刻苦 nowadays?
Yes 刻苦 is still commonly used in colloquial language.
“他那一年刻苦地学西班牙文” is grammatical. In spoken language it sounds more natural to say “他那一年学西班牙文学得很刻苦” .
刻苦 is 'hardworking' while 认真 is 'take seriously'. They are idiomatically used together ('认真刻苦'), but they are not synonyms. Actually you can say things like '他很认真，但是不够刻苦' meaning he is taking it seriously enough but he needs to devote more time.