“Everything is not…”

The correct grammatical (and also mathematical) meaning of the phrase is the former one:

Nothing is... (for the set of all things, no thing is...)

But unfortunately in the majority of cases people mean the latter one:

Not everything is... (for the set of all things, some things are not.../not all things are...)

I wouldn't say the phrase is ambiguous, it's just that people are used to using it incorrectly and you won't change that.

If you don't want to be misunderstood, don't use the phrase. If you want to annoy the people using it, misunderstand them.

Looking in Google Ngrams, people have been using the phrase "everything is not" for the last 400 years. If you actually look at the instances, virtually all of them (except those written by logicians or Buddhist philosophers) mean "not everything is", and virtually nobody uses it to mean "nothing is". This phrase is only ambiguous for logicians. The correct meaning (if you're not a logician) is "not everything is".

Unfortunately for logicians, language is not always based on logic.

Even Shakespeare had his characters use "all is not" in this way:

All that glisters is not gold;
Often have you heard that told: —(The Merchant of Venice)

All have not offended; —(Timon of Athens)

He did not mean "Nothing that glisters is gold" and "None have offended" by these.

In English, sentences containing both a negative and a quantifier, or a negative and a modal, or a modal and a quantifier, are ambiguous, unless some care is taken in phrasing them. If (as here) there is both a universal quantifier everything, and a negative particle not, then there will normally be two meanings:

• one with the negative inside the scope of the quantifier : (∀x)¬P(x)
• one with the negative outside the scope of the quantifier : ¬(∀x)P(x).

This is true of any sentence that contains any two logical Operators (Modals, Negatives, Quantifiers). In other words, such ambiguities are unavoidable, so you might as well relax; they're gonna be around a long time, and who needs the tsures?

In fact, virtually every possible English sentence is multiply ambiguous in print (though not nearly so much in speech, where rhythm and intonation usually distinguish nicely); but because we can figure out what's likely we happily ignore the unlikely though logically possible meanings, and go for the contextually sensible ones.

Peter's pertinent quotation apart, I don’t recall ever seeing or hearing, for example, Everything is not as it seems, but, whatever else it is, it’s grammatical. You don’t have to like it and you don’t have to use it, because alternatives are available. Language takes its meaning from the context in which it is used as much as from its component parts. Everything is not . . . will only rarely be ambiguous.