"The park is here."
I'm not a native speaker so I may get this wrong.
"Вот парк." means "Here is the park." but it could also be translated "Look, the park!" The word "Вот" has a meaning close to the exclamations "look!" or "behold!" in English. It can be used to mean "Here is..." but it cannot be used in a more general sense to mean "here". For this you need здесь or тут. These have a meaning more like "at this point", which is the meaning of "here" in this sentence.
But you can present it. Duo uses it in that sense in many examples. Only this time we are talking about a park instead of a country.
Your comment about the use of here would suggest that .....in this conversation the speaker is saying the park is here as if the listener was not aware or confused about where they were in relation to the park. It is possible for that to be the case but it is equally possible to point to a park on a map or from a vantage point.
Yes. In «Здесь парк», the listener doesn't know what's situated in a certain place and you tell her it's a park: Here's a park. In «Парк здесь», the listener knows about the park and you tell them its location: The park is here.
I'd say you are quite correct. With 'вот' you present something, like the French 'voilà'. It can also mean that I hand you something if it is small enough.
'здесь/тут' put empathise on the location. I admit, the difference can be rather supple sometimes. And of course one can say 'парк вот здесь' which underlines that the park is 'over here' and not around the corner, for example.
In this case, like szeraja_zhaba already explained nicely, one person just shows another where the park is situated. The other way round "here is a/the park" it could be both 'вот парк' and 'здесь парк', depending how and what you want to stress.
I believe you are focusing on the wrong word to put at the back or front, if you will. "Park" is the word dictating it's position. If the park is known, as best as I can convey, it is "The park" and you would place it at the beginning of the sentence. On the other hand, if it is just "a park", you would place it at the end of the sentence signifying it isn't a specific park. "Here" is merely the location.
Парк здесь. The park is here. Здесь есть парк. There is a park here.
I hope this helps!
here / there.
Тут is a rather context-dependent "here" associated with the place and situation speaker is in. That said, тут and здесь are quite interchangeable and both mean "here" (i.e. "in this place"): only statistics shows that different contexts prefer one or another.
Both are very common, among the top-200 most frequent words in Russian.