Difference between 是 and 有
I'm learning Mandarin from ChineseClass101.com. They teach that the verb "to be" is 是 as in 我是美国人. But the verb for existence, they say is 有, which actually means "to have," and indicated possession, as in 这儿有饭馆儿吗, "Is there a restaurant here?" (lit. "Here have a restaurant?"). But the sentence they use for "I'm here" is 我在这儿。 They mention that 在 is a preposition that means "at," so my question was if the verb for existence 有 is being omitted, then it seems like 在这儿, "at here" should be the subject and should therefore go at the beginning (i.e., 在这儿有我). They said that it was actually 是 that was being omitted as the verb "to be," which makes sense, so are the sentences 我是在这儿 and 饭馆儿是在这儿 also both correct?

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With a few exceptions (some nouns, verbs, etc) anything that says "this thing in Chinese is just like this other thing we have in English" is an oversimplification. They're two totally different languages, and there are often no direct equivalents for words and grammatical structures. Thinking that way makes things easier to understand but also will cause you to make more errors.

是 is absolutely not a direct equivalent of "to be", nor is 有 directly equivalent to "to have." they're just the closest things Chinese has.

As for the specific sentences you're asking, adding 是 seems weird. 是 can be emphatic sometimes, but the emphatic 是 is accompanied by a 的. I can't say with certainty whether it's okay or not, but I can say that it's absolutely not necessarily if all you want to do is say where you/a restaurant is.

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I think if you think of 是 as equal to, then I think you can understand that verb. Place words usually take 在. 有 can be thought of as "there is" or “”there are“” or "to have". I think if you are trying to clear up a misunderstanding, I think your last two sentences sound right. But in those sentences, 是 is used for emphases. Otherwise, it's not necessary to add 是.

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To some degree, I agree with the above answer of Seth Karp. You cannot analyze Chinese sentences by using the English structure.
I don't think when we say 我在这儿, we omit anything there. 'Sth+在+somewhere' is just the way we describe the location of something.  
When we say 这儿有饭馆儿, we use the structure 'somewhere+有+sth' to tell that you can find something somewhere. You could consider 有 as 'to exist' in this situation, but only in this situation.

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1. Having the same verb for possession and also to mean "there is/there are" is not unique to Chinese. It is the same in French and German, just to name two.
2. Just translate 在 as "to be located (at)" in your head when appropriate and...tada!

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這兒有飯館兒嗎 and 飯館兒在這兒嗎 are roughly the same in a sense, but I n the first sentence though it seems like you're already at a specific place, and are wondering if there's a restaurant there. In the second, it's more like you're talking about a specific restaurant and asking where it is.