In the following sentence, should there be a comma after the word slogan?
The 1958 Ford Edsel was advertised with the slogan "Once you've seen it, you'll never forget it."
In the following sentence, should there be a comma after the word slogan?
The 1958 Ford Edsel was advertised with the slogan "Once you've seen it, you'll never forget it."
The quotation is in apposition with slogan.
In a non-restrictive appositive, the second element parenthetically modifies the first without changing its scope and it is not crucial to the meaning of the sentence. In a restrictive appositive, the second element limits or clarifies the foregoing one in some crucial way.
If the quotation were merely a parenthetical remark, you would set it off with a comma. However, because this quotation specifies the nature of the slogan and is integral to the sentence, it is a restrictive appositive with no comma:
The 1958 Ford Edsel was advertised with the slogan “Once you've seen it, you'll never forget it.”
No. As a rule of thumb, you only use a comma around a quote when it follows or proceeds a verb that attributes the quote. You also use a comma simply when it keeps the sentence grammatical.
Lincoln said, "Better to remain silent and be thought a fool than to speak out and remove all doubt."
In this alternate phrasing, as in your example, there is no such transition:
Lincoln once said that it was "better to remain silent and be thought a fool than to speak out and remove all doubt."
Quotes aside, I'm not sure, but the way you've phrased the sentence may not be valid. Does the English language support the pattern of noun-> phrase describing that noun? You may need some kind of linking verb, in which case you would likely add a comma.