Hi kopia, being sleepy or the feeling of drowsiness is common when first starting an antidepressant. This feeling should go away soon if he has been on it for 3 weeks now. I assume he is taking it in the mornings? According to Mosby's Drug Reference Manual 2006 patients "usually" see improvement by week 4 on Paxil if this is given for Depression.
As for the side effects of stopping the medication suddenly there is a whole long list of items: Headache, nervousness, insomnia, anxiety, tremor, dizziness, fatigue, sedation, agitation, apathy, hallucinations, delusions, psychosis, vasodilatation (which is dilation of the blood vessels and can cause blood pressure problems and irregular heart beats), postural hypotension (extremely low blood pressure upon standing), and palpitations.
I would strong caution him (and anyone) not to stop this medication without physician supervision. Yes, there are a lot of people who do stop their antidepressants on their own but they feel like crap to put it nicely for several weeks until the medication has left their body.