So, I have a forgotten year of C++ under my belt and I can work with HTML and CSS. What exactly goes into web programming? I was imagining it'd be a few months studying CSS, Javascript, jQuery, Python, MySQL, PHP, Django, that Google Apps language, et cetera, and it just sounded like a lot of work.

Yeah, this kind of stuff. You will mainly need a web programming language and probably a framework. HTML is pretty simple, and CSS isn't a big deal, though it's quirky. You don't really need to know much about MySQL or Javascript to get going. (Learning database concepts will be useful later, but the skill also gives you a good payoff because it will teach you some ways of statistical thinking.) You will also eventually need some Linux.

But you only need a bit of knowledge in some of these areas to start making stuff and start having fun. HTML + CSS that you already know + programming language and maybe framework + follow very basic instructions to set up a MySQL database.

Also, I've had 2 occasions where I spent a few hours looking for resources to help with a problem and I just had to give up, which is hella frustrating.

Yes, this will happen in programming. Sometimes a session of programming will be like banging your head against a brick wall until it breaks, then moving over a few inches and banging your head against another spot in the wall.

Documentation + Google is your friend. Most problems I run into that I can't immediately solve myself can be solved within a few minutes of Googling. Of course, it helps when it's typical web programming problems that already have like a million people on the internet asking the same question as you. Just try something, Google the error message, try what it says, then Google the next error message until there are no more errors and things are working. I can do things I've never done before very fast that way.

Many problems will take several hours to solve. Some will take days to solve. When you spend all that time stuck on exactly the same thing, then it's frustrating, especially if when you solve it, it turns out to be something stupid like a typo or a badly documented quirk of a function or something. For me, that's the exception, not the rule. Even when working on a tough problem that takes a long time to solve, I'm typically making progress along the way. When the problem turns out to not be something stupid or obvious that I missed, then it's a really nice feeling of accomplishment when you solve it. Variable reinforcement schedule, and all that. It's a great feeling to finally solve a challenging problem that took you days, when you were never even sure if you could solve it.

The result is that how you feel about programming depends a lot on where you are in your current problem-solving cycle.

I dunno, it feels like there's this big unknown gap between me and being a below-average web programmer, and I don't like being below-average at anything, let alone having to work hard to become below-average.

I think it's possible that you've been a bit too hard on yourself, and not given yourself enough time to see what programming is like and assess your aptitude and enjoyment for it.