the tappets are to make sure the valves can positively close off and compensate for expansion, It does not matter if the fuel induction is carburettor or fuel injection.
Your car not having a temperature guage is a silly move by the manufacturer, You can see the engine temperature if you connect some sort of scan unit to the OBD port under the dash or use an infrared thermometer on the engine itself. If its heating up quick, your cooling fans will also start early.
Is that hapening?