I don't see any reason to use quotation marks at all in your example sentences; you don't appear to be quoting something someone has said, and restaurant/convenience store names are not inherently written with quotation marks. In addition, the apostrophe is not appropriate in your second example; multiple 7-11s are plural, not possessive, and so you simply add the s without an apostrophe.

That said, here's an example to illustrate how you might use apostrophes and quotation marks together in a sentence where it makes sense:

We looked at the signs above each of the diners along the street; the one that read "Mary's" looked most inviting.

Note that you don't have the problem of the plural in this one, so the punctuation is rather simple. I'm trying to think of a situation in which you would actually have to use quotes and an 's at the same time, and one doesn't come to mind. If you're quoting speech, it's unlikely that you're going to pluralize the whole quote. And if you're saying "There are five McDonald's in my city", you're not going to put McDonald's in quotes, and it's debatable if you'd just leave McDonald's as-is even though it's plural, or try to write the unwieldy McDonald'ses (which I wouldn't). So if you can think of a situation in which this would actually occur, I'll try and update my answer to reflect. But nothing comes to mind as a situation in which you would need to do what you're suggesting.

I don't see any reason to use quotation marks at all in your example sentences; you don't appear to be quoting something someone has said, and restaurant/convenience store names are not inherently written with quotation marks. In addition, the apostrophe is not appropriate in your second example; multiple 7-11s are plural, not possessive, and so you simply add the s without an apostrophe.

That said, here's an example to illustrate how you might use apostrophes and quotation marks together in a sentence where it makes sense:

We looked at the signs above each of the diners along the street; the one that read "Travis'" looked most inviting.

So yes, in this case you would place an apostrophe before a closing quotation mark. You're directly quoting the sign, so there's really no other option. It looks a bit odd, sure, so you might try to avoid it, but it's correct. Additionally if you use a font that distinguishes more clearly between apostrophes and quotes, it might be easier to read.