Exactly. I don’t see what the big deal is.
Yeah, the point that “its” (no apostrophe) is a pronoun that indicates possessiveness (that phrase sound familiar?). Why do you think “its” should take an apostrophe when “his,” “hers,” “yours,” “theirs,” and “ours” do not? The whole point is that it IS a pronoun, and unlike common nouns and proper nouns (like your “John” and “Mary” examples), POSSESSIVE PRONOUNS DO NOT USE APOSTROPHES. NONE OF THEM. NO EXCEPTION FOR “ITS.”
Out of curiosity, if you were to substitute “it” for “me” in “The woman next to me’s dress is beautiful”, how would you spell it?
Misread the question. Ignore this post.
With an apostrophe. Here we’re not dealing with a possessive pronoun, rather with a phrase (“woman next to it”) that happens to end in a pronoun. Different animal.
That sounds more like a question of phrasing.
“The woman next to me is wearing a beautiful dress.”
Ah, fair enough. That would, I think, be my preference as well, and would mesh with the reasoning you took. However, others in this thread have, at least implicitly, stated the opposite preference for that situation, so I was curious.
I don’t want to rephrase it; I’m perfectly happy with the phrasing I gave. I’m only curious how others spell the phrasing I was asking about (“The woman standing next to it(’)s dress is beautiful”).
Not quite. “It’s” can also stand for “it has” (e.g. “It’s been a lovely evening”).
I can see that opposite preference leading to trouble and confusion.
“See that big statue?”
“Yeah.”
“The woman next to its dress is the same color as my new purse.”
“Wait – the statue has a dress? And your purse is the color of her flesh?”
Make it “The woman next to it’s…” and the confusion is avoided. In print of course – it’s hard to hear those apostrophes in speech. 
But using “me” or “him” instead of “it” illustrates the right way to do it. “Woman next to me’s…” or “woman next to him’s…” make it clear that the pronoun is in the objective case as the object of “to,” and that the apostrophe-s must apply to the phrase as a whole. One certainly wouldn’t try to say “next to mys/my’s…” or “next to his’s…” – those would both change the meaning to nonsense and present a word that doesn’t exist. Likewise, “next to it’s…” takes “it” as an object of “to” plus apostrophe-s to make a possessive of the phrase. Using “its” would just add confusion and screw up the meaning.
The one that gets me most is discreet vs. discrete. It’s not that hard, and the meanings are completely different!
Agreeing text is in agreement.
Unless the apostrophe was being used as a contraction, as in a bakery calling itself the Don’t Shop. 
If you don’t know the difference, whenever you need to use one or the other always choose the other.
On another message board I’m on, we’ve got a guy who has problems with knew/new. He’s the only person I’ve seen with that particular issue.
It seems particular grammar mistakes run rampant on certain boards - like someone said earlier, monkey see, monkey do. On that board, there’s a serious problem with fragment sentances. (For example, “Bob walked over to the table. Because that’s where the mail was.” or, “We should do A instead of B. Since it’s better.”). Over the years (it’s a theatre group I’ve been involved with for about six years), you can see the evolution. It started with one guy who has terrible grammar, but posts a lot, to a group of about five folks who have slowly regressed in their ability to form a complete sentance.
Except for “one’s”.
The phrasing that you gave, though syntactically possible, is highly awkward and I would not permit it to remain in a document I proofread. I would say “The woman next to it is wearing a beautiful dress.”
You’re right, good catch. I stand corrected.