Do all LPG gas appliances need a regulator?

There needs to be a regulator to keep the pressure at the appropriate level. The tank may have an adequate regulator but make sure the pressure it allows is compatible with the appliance. 

No.  The regulator is on the propane tank, underneath that dome cap on top of the tank.  If your tank is say 50 feet away from your house (and it should be), there are 2 regulators.  The first stage regulator is on the tank and releases propane at a higher pressure into the tubing line to the house (say 5 or 10 psi?).  Then at the house, there is a second regulator that drops the pressure to that required to appliances (a few ounces).  Talk to a propane dealer about the pressures and regulators. You want to use only certified equipment from a certified propane dealer.  The propane then goes into the house at the conventional pressure to service all of the appliances, similar to natural gas service.  Our house  runs a 1" steel pipe to the gas furnace area at the far end of the regulator, and then 1/2" steel pipe off of the header to gas appliances.  Longest run is about 100 feet from the regulator and has no problem with supply.  Propane tank is about 200 feet from the furtherest appliance, a tankless water heater.   

A gas regulator is used to shut off the gas if the pressure goes too high. A gas fired boiler has two regulators. one if the gas pressure goes too low.

Yes.