J.M.
I don't really have any good advice for your first question, but I usually try to avoid giving really small kiddos adult medication. Have you asked a doctor? Cool washcloths on the forehead, wrists, or neck can sometimes work.
The second part of your question is easier. When a child says they are sick and you suspect that they are not, treat them as if they are *very* sick. Make them lie down - preferably in a quiet, dim room away from activity. Feed them food that is as nutritious and bland as possible. Don't let them do anything that could get them too excited, like playing video games. Baby them. In short, bore them silly while wondering out loud if they will need to go to the doctor. If they aren't really sick, they will tire of this treatment very quickly. If they really are sick, then you're doing just about everything you can for them. When they say that they are well, keep up the act a little while longer. After all, you wouldn't want them to "relapse" and get ill again!