Re: Help -
Welcome to the garage.
When you put a new battery in after the other went dead, it can happen that the PCM has to relearn the idle settings. Try to keep the engine running using the accelerator (while standing still), until it heats up. Normally it will be ok then Do you know why the battery was dead? A bad alternator could cause the engine problem and tje dead battery. Check battery voltage with engine off, should be around 12.7V with full battery. Then start the engine and see of the voltage went up to 13.5-14.5V . If it didn't change, you probably have a bad alternator. You could remove the alternator lead (with engine off and isolate the lead to not cause a short) Now testdrive and see if the engine runs ok..
Wet floor and anti freeze smell is most probably the heater core. Whole dash needs to be removed, to get the HVAC housing out.
The leak at the radiator cap is most probably the cap itself.