No heat is often an indicator of either a bad water pump, stuck thermostat, or dangerously low coolant level, which you say you don't have. Thermo is easy to test and easier to replace. If it's faulty, go with an OEM part. Also verify that the housing for the thermostat isn't leaking. I highly doubt that's the issue (keep reading) but it's possible. Start with testing the small things then work your way up.

Additionally, you need to check for a blown headgasket. Looking at the spray pattern on the resonator, coupled with the coolant you are seeing in the battery tray, leads me to believe that coolant is likely coming out of the overflow reservoir that is being pressurized by combustion blow by getting into the water jacket. I've seen it before. After the car is warmed up, can you see coolant up to the neck in the overflow tank? This is often a telltale sign.

 

Judging by the color of the coolant, you may have mixed store-bought generic coolant with the Honda factory coolant, which can lead to clogging due to chemical differences between the Honda factory coolant and the store-bought stuff. Isolating the heater core and flushing it may show you that there's a bunch of gunk in the heater core blocking the flow.

However, due to the sheer frequency of blown headgaskets, its not unlikely you already had a blown headgasket.

 

I am going through this with my sons 03 civic and I have determined that whats happening is that the head gasket has a small leak and is causing an air pocket in the coolant system. After doing research it seems to be a common problem with these 1.7 engines. Wish I had known this before buying the car. Now I am trying to figure out if its worth my time and energy to just slap a new gasket on it or sell it as is.

 

no heat usually means there's a blockage in the heater core