"Is this a menu?"
It's to extend the length of the vowel sound it comes after.
As it is now, it would be pronounced "menyuu", but without the ー it would be pronounced "menyu".
I believe it's only used in katakana, but I could be wrong.
In Hiragana, they usually write the vowel instead of using the dash –. For example, "menyuu" –> "めにゅう". (BTW, this as a foreign word is very strange see it written in Hiragana).
Correct, it is really only used in katalana. In hiragana, vowel sounds are usually extended with the う regardless of the previous vowel like in sayounara (さようなら) where you have a u after an o sound, but you dont emphasize the difference in sound.
That's not quite right. You use う to extend the 'o' sound, but if you followed any vowel sound except for お or う with う, you'd change the sound rather than just extending it. え is usually extended with い, which also extends itself and あ is used to extend its own vowel sound.
There's nothing wrong with it; 「これはメニューですか」 and 「メニューはこれですか」 are largely interchangeable in most circumstances, but their nuances are subtly different.
With 「これはメニューですか」, これ is the focus of the sentence, so you're asking if "this thing" that you are holding or pointing to is a menu (and not something else). That is, "Is this a/the menu?" As such, I presume you're more likely to hear and use this sentence.
With 「メニューはこれですか」, メニュー is the focus. So this time you're asking if the menu is "this thing" that you're holding or pointing to (and not some other thing). That is, "Is the menu this one?"
When the restaurant doesn't give the same menu as you expected it to
Earlier we learnt that "re" in "kore" means the noun isn't being mentioned. I thought this sentence would use nore instead, since the noun menu is mentioned? のれはメニューですか
I'm not entirely sure what you're trying to say but you see confused. これ simply means "this".
I'm pretty sure saying "meniyuuarimasuka" would make the sentence translate into "I have a menu?"or "Do I have a menu?"
Kinda. メニューがありますか literally translates as "is there a menu?" but from context could be either "do I have a menu?" or "do you have a menu?"