The rules are the same in Czech and in English:
The Earth's moon is called "Moon" in English, "Měsíc" in Czech. Our sun is called "Sun" in English, "Slunce" in Czech. Other moons of other planets and other suns in other solar systems are still moons and suns, but they have different names, not "Moon/Sun". This planet is called "Earth" in English, "Země" in Czech. Lower-case "earth" means dirt/soil and lower-case "země" means ground/soil.
That said, it's not mandatory to capitalize the Moon/Měsíc or Sun/Slunce, unless you're talking about astronomy.
And adding "that" is just adding emphasis - you point at the moon and exclaim "Wow, look at that moon - how big it is tonight!"