Can my 4 year old take cough medicine and tamiflu (oseltamivir) at the same time? Positive influenza test 4 days ago, lingering cough, fever free 48 hours. Contagious?

Yes, most. over-the-counter children's cough medications can be taken with Tamiflu, (oseltamivir) but the cough medication may not have any added benefit and will not shorten the course. If he/she is still coughing, I would consider him/her to still be somewhat contagious through those respiratory droplets released by the cough, although one is most contagious while the fever is there.
Flu and cough. yes. He can take both together. He will secret the virus for sometime.