Yes! If you're not a fan of the way the brim is written, you can really easily substitute any other brim you prefer. I would recommend that you still work the brim "sideways" in short rows and then crochet along the edge for the body of the hat in order to maintain the right stretchiness that you would want for a brim. Using something like a fpdc, bpdc pattern wouldn't have the same elasticity, even if it would have a similar ribbed look.
This is the round where you make the ribbing by alternating between sc and fpdc. If you work both a sc and a fpdc in the same stitch, then this would double the number of stitches. You do NOT want to do this. You want to make sure that you're only working one stitch in each of the stitches from the previous round so that you can maintain the same number of stitches. In other words, in the first stitch, you'll work a sc. In the second stitch, you'll work a fpdc. In the third stitch you'll work a sc...and so on.
For round 12, should I work the fpdc around the sc in round 11 or around the fpdc from round 10?
You should work the fpdc from round 12 around the fpdc from round 10. That way the posts of the fpdc will be connected, making a neat ribbing. If you work around the sc you'll get more of a gap, which could also be another explanation for for the next problem (see below).
My ribbing doesn't look right; there's a gap in the middle of it. What's wrong?
This gap can be caused by the round of sc (round 11). You'll notice that on mine, it looks neat and connected. But if you're a really loose crocheter or if you make really tall sc, then you might get a small gap. In this scenario, you can just skip round 11. In the picture below, I've done the first bit of ribbing (at the bottom) as written and the second bit of ribbing (at the top) omitting round 11. As you can see, they look fairly similar. I find that without the round of sc, the ribbing tends to lean a tiny bit to the right and the dc are more scrunched looking, but it's barely noticeable (and may not happen if you're a loose crocheter). So I still recommend round 11 if it doesn't disrupt your ribbing because I think it looks neater, but if it does make a gap, just skip it. It should be fine.
This can happen if you join the yarn around the brim with the wrong side facing out. The pattern specifies that the right side should be facing out as you begin to work in the round in order to avoid this. But, life happens sometimes, so don't panic if this happens to your hat. Once you start working rounds 4-8, it'll become really obvious which side is facing out. If the wrong side is facing out and you don't feel like frogging it, you can just changed the ribbing. You can either do a [sc, bpdc] around, or you can do a [fpdc, bpdc] pattern, which will be reversible. Then you can just flip the hat inside out when you're done. It won't look exactly the same because the "wrong side" of sc stitches will be facing out, but it'll still be a really similar looking hat.
I think my hat is inside out. When I work the body of the hat, the 3rd loop is in the back. Why isn't it in the front?
Okay, so this actually isn't a problem; it's supposed to be this way. When you work in the 3rd loop while making the brim, the 3rd loop will always be in the front, but that's just because you're turning the piece as you go. When you work the hat in the round without turning, the 3rd loop will always stay on the inside of the hat. I tried to get a picture of working the 3rd loop on the backside (see the picture below), but if this still seems confusing, I recommend working the snail pattern from the 3rd Loop Tutorial. Either way, your hat is correct, so keep on going!
There are basically 3 different scenarios here. Depending on which way your hat is "too big," you'll want to do different things to fix it.
Scenario 1: The hat is too big all around
This issue comes down to gauge. Did you make your gauge swatch? (Instructions are found in the original post.) I know, I know. No one likes making gauge swatches. I'm guilty of ignoring them too. But would you rather spend a few minutes making a small square or spend hours making a hat, only to find it's way too big? As a show of solidarity, I made one with you. Feel better?
Scenario 2: The circumference of the hat is too wide
If you're happy with your gauge, but you just want a different circumference for the hat (if you're making a child sized hat, or if you prefer a less baggy slouchy hat, for example), then you can just decrease the number of stitches for the body of the hat. As it says in the pattern, as long as you have an even number of stitches, your hat will work out great! This gives you a lot of flexibility to personalize the hat however you'd like it to fit. In case you're wondering what the original dimensions are when following the gauge as written, the original hat is approximately 20.5" in circumference at the widest point (the measuring tape slipped a tiny bit in the picture below).
If you prefer a less "slouchy" slouchy hat or if you just want it shorter (as for a child sized hat), you can decrease the number of rounds in the hat. If it needs to be just a little bit shorter, I would recommend omitting some of the final hdc rows. If you need a couple inches or more off the hat, you can alter the pattern by omitting the last repeat or by decreasing the number of rows in the ribbing or 3rd loop section. If you'd like an idea of how to do this, the pattern for the messy bun version of this hat is a great example. And, if you're wondering about the original dimensions as written, the original hat is 10" tall (including the brim) before closing up the top.
Closing off the hat is actually a lot easier than it may look when you're just reading it. When you actually have the hook in hand, it should seem a lot simpler. But if it doesn't, never fear. There's a basically foolproof alternative. After you finish crocheting, fasten off the hat and leave a long yarn tail. Then, using a tapestry needle, weave the tail through the last round. I recommend weaving it through the front loop only of each stitch (as shown below), for a tighter closure.