The ð is the soft d. How to pronounce the word would depend on dialect. Some people would include the soft d sounding "goðdaj" with an equal emphasis on both syllables. This would be what you would hear in the royal theater or in old Danish movies as this indeed used to be a common pronounciation among the better classes of Copenhagen. In more modern speech it would more common just to say "goda" with emphasis on the last syllable. In Jutland it might be more common with "godau".
the phonetics (without audio) from Den Danske Ordborg:
It is correct that in the language from hundreds of years ago the influence from English is limited, although some dialects along the west coast of Jutland are very similar to dialects on the other side of the North Sea, so that mutual influence is the likely cause. However, in maritime language there are words that must be a few hundred years old in Danish and in modern Danish there are tons of English words - and even sentence constructions.
A lot of languages have words sounding very similar to English. They're cognates. I know many languages have them as well, but as an example, Japanese has 2 alphabets and the Chinese kanji system.
The first alphabet is the first you learn, in like 1st grade of elementary school, and the second alphabet you might learn in 2nd grade of elementary school. The first (Hiragana ひらがな) is solely for writing originally Japanese words, or writing the pronunciations of the kanji. The second (Katankana カタカナ) is for writing foreign words in Japanese, and is also used to write most animal names. Japanese has a whole alphabet, just for foreign words.
Yes, Good morning/bom dia should also be accepted - as well as good afternoon/boa tarde. In English Good day - is something you might say when leaving, as a statement that now we are done. You could never use goddag in this way in Danish. There is a separate phrase "godmorgen" in Danish, but this is reserved for early morning. If you use it after 10 am it could be ironic to indicate the person is very late to arrive this morning.
I have never said "Good day" as an English person. Some people below talk about this being a kind of farewell in English, but I have never heard it used in this way either. To us Brits, "Good day" is only used by Australians; we would always specify, "Good morning/ afternoon/evening", or else use "How do you do?" (fairly formal), or "Hello/hi" (informal).