Personally, for many years, I preferred and used ". and not ." because I thought, "The period ends the sentence and it should go last." But when I started editing for scientific journals, were details are important, I found that the second way is correct, "Like this," I might say, "Or like this."  I no longer use "this method".

The quotation mark use could be considered style but here's my thoughts.  I am guessing, but I suspect that a search of the internet would confirm my ideas.

"Double quotes" are more formal.

'Single quotes' are less formal.

AND I know for sure.

"You should use single quotes for a 'quotation within a quotation' as you see in this sentence."

That is, I would use double quotes in most cases, such as quoting a person, "Like me?" he asked.

I would also use double quotes for quoting the titles of books such as "The Sands of Mars" by Arthur C. Clarke, who I call 'my favorite author' less formally.  About 20 years ago, it became 'popular' to put 'quotation marks' around almost anything, so I might use 'single quotes' in that way.

I did search the internet briefly and learned the Brits tend to use more single quotes (quotation marks) and North Americans use double, so it is a matter of style, as I suspected.

Just don't get too "carried away" by using "too many" "quotation marks" like this sentence that exaggerates what I've seen in "recent years" far "too often."

pH