1) Check to see if your plug is fouled, it is starting so you are getting some gas and a weak spark. Clean out the spark plug tip with a wire brish or dremel tool. If it starts better, spend a few bucks on a new spark plug.
2) If your lawn mower is old or it has been outside all winter, spray the carb with carb cleaner to losen up all the gum. You might not be getting a good gas / air mixture.
5) Unlikely, but it could be old gas, or if it is a 2 stroke where you have to mix the oil into the gas you may have too much oil in the gas.