"Where am I?"
It means this. Remember the こ、そ、あ、ど series: こ refers to something near the speaker, そ refers to something near the one being spoken to, あ(そ) refers to something not near any participants in the conversation, and ど is inquisitive.
So, [ここはどこですか] is asking where is here, literally.
I don't think it's technically incorrect, but it's definitely not what Japanese people would say in the same situation. Remember, they tend to avoid using 私 if they can and ここ does that quite neatly.
To me, 私はどこですか actually sounds a bit more literal, like "where is me?" which you might say if you lost track of your physical body or if you looked in a mirror and couldn't see yourself.
Since どこですか doesn't supply a subject to the sentence, so it really just means "Where is it?" or "Where?" If a subject was already given earlier in a conversation, for example "そのいぬはかわいかったでした" (that dog was cute), and someone responded "どこですか?" they'd be asking "where is it?" どこですか is especially wrong in this context since you can't really ask "where am I" without stating the subject, being "I," so you'd use "わたしはどこですか," which is specifically asking where you are.
Not quite, I think it's simply ungrammatical.
Instead of thinking of です as "is" or even "to be", I tend to think of it as a kind of equals sign because it avoids the confusion of "is" being used for "existence" in English. です works by equating the subject, indicated by は, to the object which comes right before です.
Here's a couple of simple examples from earlier in the course:
・(私は)アメリカ人です the implicit subject, "I" = "American person"
・(あなたは)イギリス出身ですか？ the implicit subject, "you" = "originate from England"?
・それは何ですか？ "it" = "what"?
As you can see, the exercise here follows the third pattern, becoming "here/this place" = "where"?
You asked if by leaving out どこ, it becomes "is this real" (which actually follows the second pattern, "this place" = "real"?), but by leaving it out, you leave the equals sign hanging which essentially makes it ungrammatical. (Note: "does this place exist" no longer has "is/am/are" as the main verb, so you would have to use a different verb from です.)
The gap in the equation can theoretically be filled in by context, but at this point absolutely anything goes. ここはですか sounds like you're confirming the implicit subject of someone else's sentence, where they gave only the object.
I don't think it's technically incorrect, but it's definitely not what Japanese people would say in the same situation. The implied subject of that sentence is almost always a third party; "where are you", "where is it", etc.
To me, if the implied subject was 私は, it would actually sound a bit more literal, like "where is me?" which you might say if you lost track of your physical body or if you looked in a mirror and couldn't see yourself.