In the first sentence of a story, is it okay to only refer to the POV character with a pronoun? For example: He was having the worst day. VS Mike was having the worst day. In first person, you usually don't find out the character's name right away. Would that work in third?
It's grammatically non-standard (not 'supposed' to use a pronoun without an antecedent) but grammar rules are broken all the time in creative writing. If it works, it works. But make sure you have a reason for it, and aren't just being sloppy.
I think you're talking about something different. Pronoun-antecedent agreement refers to the pronoun referring back to some noun such that there's number and person (1st person, etc.) agreement between the two. For example, consider The word "his" agrees with "the boy" in both number and person. Now take something like this: Putting aside issues of common usage, the word "their" (plural) doesn't agree with "everyone" (singular) in number, so it's more appropriate to write it like this:
I'd say also, that there's a potential impersonalization that occurs, at least regarding norms. We think of people by names. When we can only think of a person by a pronoun, he/she becomes a vague "thing." He/she is less individualized, more of the piece. Just my take. Not saying that you shouldn't do it. You may be trying to create this distant atmosphere. Just depends on your goals. If you don't have a goal in this regard, then I'd probably follow convention. Edited at add: In terms of your two examples, I'd say that the first (pronoun) creates a setting that is more universal. The latter (Mike) creates a perspective.
Well, you understand what an antecedent is, yes. But a lack of agreement between pronoun and antecedent is only one of the problems one can run into. In this case I'm not talking about the pronoun not agreeing with the antecedent, I'm talking about the antecedent not EXISTING. Which is also a problem, in terms of conventional grammar rules. You could read "Problem Number 2" at http://www.quickanddirtytips.com/education/grammar/pronouns-and-antecedents?page=1 for more explanation, if you need it.
Then what you're referring to isn't an antecedent problem. It's something called a cataphoric (or anaphoric for the opposite case) pronoun. Anyway, this isn't a problem, either, and I'm honestly not sure why Grammar Girl even bothered mentioning it (she's usually a very good reference). Looking at it from a descriptive point of view, using cataphoric pronouns is purely a stylistic preference. If you look at it from a prescriptive point of view, it's still not a problem. The CMoS allows for this. No other style guides I've come across say anything about cataphoric pronouns despite how common they are, so it's safe to assume that using them is fine.
I didn't say it was an antecedent problem. How could it be an antecedent problem when there's no antecedent? It's a pronoun issue. (And in my original post I said I didn't think it really was a problem, just non-standard. And I stand by that. Pronouns are 'supposed' to be clearly linked to nouns, and I see no linkage here.) If you're talking about cataphora (I googled "cataphoric pronoun" and there were less than 2 000 mentions on the whole internet, so... I think the reason you're not seeing it mentioned in grammar guides is that it's a really obscure phrase), then I guess you're assuming there's a postcedent somewhere later in the writing. Fair enough. I was assuming that if there was one, it was an unreasonable distance away. Neither assumption is testable, really. Regardless, I think we agree that there's no actual problem in using this structure in creative writing, which is what the OP was asking. So... yay! We agree!
But why does the pronoun, 'he', need to be identified by name in any work of fiction? A lot of stories are about unnamed persons. I looked over your link and I don't find it the best advice for fiction. Perhaps it refers more to nonfiction. I get where the Grammar Girl is coming from, but I think the target audience there is basic writing. You see a lot of inexperienced writers messing up pronouns.
The pronoun 'he' DOESN'T have to be identified by name in any work of fiction. That's what I said in my post. A pronoun with antecedent isn't grammatically standard, but it's just fine for creative writing.
Thanks, everyone. I'm still mulling around different sentences. I found that if I used a pronoun, my sentences were more creative. I think I had a block from trying to do everything correct. Same thing happens when I try to fit in the where, what, and still have a hook.