Signs, signs, everywhere there's signs

Having in recent years taken an undergraduate degree in history, at a British university, I have had a level of instruction on punctuation, footnoting etc. Below is how I would punctuate your examples.

But let me say from the outset that I am assuming that whether the 'sign' contains a full stop is totally irrelevant to the reader. If it were a matter of great relevance, perhaps a forensic clue to a major crime, I would add footnotes to explain. I certainly wouldn't rely on the reader picking up the correct inference from the subtlety of the punctuation!

The sign said 'Handle with care'.

The sign said 'Handle with care'.

Pete said, 'The sign said "Handle with care" '.

Pete said, 'The sign said "Handle with care"'. [This most definitely has to have the full stop last. The only instance in which I would not put the stop last would be if the whole thing was in quotes.]

"The sign said 'Handle with care'", Mike said. [I would put the comma after the second close quote though I wouldn't be dogmatic about it.]