If you feel Dobrý den! is not the right translation of Hello! please read the discussion for the reverse direction https://forum.duolingo.com/comment/24325734 first. The translation was selected carefully considering several important aspects. Also note that "Good day!" is accepted, just not selected as the primary translation.
This is true. But learners should keep in mind that ahoj is an informal alternative, more like "hi/bye" than "hello/goodbye." The latter are more universal, meaning you can use their Czech equivalents (Dobrý den/Na shledanou) in pretty much any context without worrying about possibly offending someone. There's still a little more expectation of politeness in certain situations in the Czech Republic than, say, in the US.
Den is a masculine inanimate noun, like obchod in the first table from your linked site. So it needs the correct form of the adjective for a masculine inanimate noun. The standard (non-colloquial) form in this case is dobrý. Dobré is the standard (non-colloquial) form for neuter nouns.