There is one inlet and outlet at the engine oil cooler. Are they for the blue coolant running through to cool down the engine oil?

Also, they are marked as E-15-10. What does it mean?

 

The "E" lettering is to send the Honda parts guys to another section for the coolant hoses that run to and from the cooler.

It's both a oil warmer and oil cooler. When the engine is warming up, with cold oil, the warm coolant speeds up bringing the oil up to temperature, and when oil temp is higher than coolant temp, the heat is transferred from the oil to the cooling system.

Since it's a fairy small heat exchanger, and the cooling system isn't the largest out there, in severe use, when oil temps are really high, an external air/air or dedicated oil/water cooler system should be used, with a thermostat system to keep temps at minimum level. You can over cool oil.