When should I bring my 2-3 year old child to the doctors if I suspect that my child is sick?

Agree. I would add 5) if your child is difficult to arouse (by this i mean hard to wake up, not lying on the couch like a lump because he doesn't feel well), and 6) if something specific hurts (like an ear or throat), or 7) he refuses to drink or hasn't passed urine in many hours.
Four signs. If your baby 1) is fretful or irritable for more than 24 hours, 2) has fever that is lasting more than 48 hours, 3) has temperature higher than 103.5 degrees rectally, or 4) screams and cannot be calmed down in 3 hours.