Did you buy the car with verified auction sheet?
Regarding fuel average, try to get your car tuned from Toyota 3s. Get the throttle body cleaned, plugs checked, filters cleaned and also computer check up.
Late start is due to battery. In winters the weak battery usually shows symptoms. So get the battery changed.
The vibration is normal in 3 cylinder car. The tuning will improve it.
And regarding jerks and missing I guess your car needs a fuel pump replacement.
Lastly use hioctane. No matter what. It really makes a difference in performance and mileage.