Yes, they are grammatically correct.
While I wouldn't call it common, it's not at all obscure a construction. Making them up off the top of my head:
(After seeing the HR department stuffing pink slips in envelopes, the layoffs were) not that much (of) a surprise.
(Being told she looked like his mother was) not that charming (of) a compliment.
("Buy five get one free" is) not that much (of) a discount.
(And it's) not that great (of) a deal (either).
(He's) not that fluent (of) an English speaker.
(He's a fine programmer, but he's) not that much (of) a sys admin.
(emacs is a fine operating system, but it's) not that great (of) a text editor.
(After last season's use of the fashion, it's) not that fresh (of) an idea.
In almost all cases, it's used to connote disappointment. It's a way of saying that something did not meet a standard, particularly one that seems self-evidently pertinent because it has to do with proposition of what the thing is in the first place: an inadequate sale can barely be said to be a sale at all. If I can get a can of peas at store X for $0.50, the fact that store Y has them 20% off their regular price of $1.00, i.e. $0.80, will make me think, "Well, store Y, I suppose it is technically a sale, but I'm having trouble going along with your proposition that it's a sale to the point of thinking of it as one, because the savings of this sale are so inadequate to compete with store X that, it's not that much of a sale."
It's only very slightly informal, but I can't think of a way to use the expression of disappointment in something (which this almost always is, unless used ironically) that doesn't risk offending someone. You'll want to be careful using it, constraining yourself to employing where you really do mean to let someone know that something is inadequate, and when you're prepared for the sort of reaction that can provoke.
Interestingly, you have the classic exception at the top of your list: "Not that big of a deal", which is used to reassure that something hasn't given offense or caused distress.
In American English, the idiomatic wording of your sentence when including the time is:
I'm running 5 minutes late.
Note that most style guides suggest spelling out numbers less than ten, thus:
I'm running five minutes late.
Best Answer
All of those phrases are wrong, as it happens.
The correct phrase would normally be "[I am] sorry [that] I am late", but you could also say "[I am] sorry for my lateness". (The words in brackets are optional, but implied if you leave them out — that's just the way the sentences are constructed. And of course you can contract "I am" as "I'm" as usual.)
You can't be sorry (an adjective) for late (an adjective); you can be sorry for lateness (the noun form of that adjective), or for being late, or you can apologize for being late, or for lateness, or whatever. But if you did something a little wrong (being late), you're the one who would apologize for that, not the others. Instead, you could (if you wanted to be rather formal) say "[Please] forgive my lateness", or even "[Please] forgive my delay". But on an MMO that formality probably belongs in in-character/RP chat and nowhere else.