the water mark left on the ground or around your engine components is probably condensation/moisture, that could come from your A/C. That is normal - most cars do it.
Your radiator fluid should be checked when your car is cool. Not hot. The overflow reservoir will give the most accurate reading when it's cool and it's nearly full.
You've probably already had a quick read through this section, but if not, review item #2 "FILL ENGINE COOLANT" section.https://celicahobby.com/forums/...ct-replace-or-drain-fill.html#Post296224
Coming from a very low/to none coolant level, the anti-freeze filling requires a gravity-type of system to fill. The instructions for that are outlines in the link.
A few other ideas of things to look at, on top of the anti-freeze fluid level/correct fluid type, is the temperature gauge functioning correctly (which you mentioned above). When it's HOT, it will be near or just above 195F, and will display as mid/upper-mid level on your dash/gauge cluster for that temp. The radiator fan should engage at about 199F.
If you have an ODBII reader, you can pull the codes that your vehicle is throwing. Or if you have an ODBII Bluetooth dongle, you could monitor the temperatures with your Smartphone or Laptop (I use the "Torque" app with a custom plug-in on Android) which your Celica's computer system (ECU) believes your engine is running at, in real time. For example, see attached image: