"Latin alphabet" is ambiguous, you understood this to mean "the classical Latin alphabet", I was sloppy and meant "the set of letters in the Latin alphabets that make up the Latin script". Now it's my turn to call you out for a very strained sense because in contrast "Latin script" is well defined and I see you are straying away from the definition for some reason.
ø, ç, ß and ő are definitely part of the Latin script.¹ Any Latin letter was added to the set at some point in time; the definition makes no special distinction for this fact.
> here's Russian Cyrillic for "restaurant": PECTOPAH
That's not how scripts work. Those are Latin look-alikes (homoglyphs). These are the correct letters: РЕСТОРАН
Just because the letters have a common ancestor, it does not mean they are the same today.
> a way in which Russian writing differs from Latin script, but Danish writing doesn't?
Russian uses the Cyrillic script (specifically Cyrillic letters from the Russian alphabet), Danish uses the Latin script.² The scripts are distinct sets. It's quite tautological when I write it that way, but I don't know how else I can makeself understood.
¹ In a technical sense, we are constrained here on this Web site by communicating within the confines of Unicode. You could look up the properties of "ø" and see that it is indeed a letter in the Latin script: `\p{General_Category=Letter}` `\p{Script=Latin}` Currently, there are 1335 registered in this set product. The properties don't come from nowhere: Unicode merely codifies what was already linguistically/sociologically agreed upon beforehand. IMO the standard is not quite as expressive as e.g. me sitting next to you with pencil and paper, but good enough for most practical purposes.
² As always, there are exceptions for niche uses and because human language is a messy concept, but we can ignore that and concentrate on the broad strokes. An example for an exception would be that names mentioned in a Latin script embedding are typically transliterated/adapted instead of remaining in Cyrillic, e.g. "Puschkin"/"Puškins"; that's a Russian word, but written in Latin.
This is only true in a very strained sense of "the Latin alphabet". No two of the languages you mention have an alphabet in common.
The Latin alphabet has no U or J, and while Y and Z were known to Latin speakers they do not appear in any Latin words.
Compare ø, ç , ß, and ő.
Even alphabets that appear visually similar may be quite different. The Spanish alphabet I was taught went:
A B C CH D E F G H I J K L LL M N Ñ O P Q R RR S T U V W X Y Z
The Ñ is not the only difference there.
Something analogous is true for the different varieties of Chinese; there are a handful of "regional" characters with no standard use.