Can touching and poking a lymph node in armpit cause it to swell and become painful?

Yes indeed: If you poke often enough and firmly enough, the area will definitely become tender and probably swollen. If you can't feel an obvious large lump by very light pressure, then nothing is wrong -- don't keep trying. Anyway, there are very few illnesses that cause lymph node enlargement as the only symptom. If you otherwise feel well, there's no point in looking for lymph nodes.