To address a couple things others said:
There's not a whole lot of difference when you're comparing the teenage years, so you will have to keep their maturity in check (while also staying consistently believable, there are very few thirteen-year-olds with a good grasp on how the world works). The same goes for characters in their 20s and up.
I have to take some issue with this. It's certainly true of characters in their early twenties that "journey fic" about them won't differ tremendously from journey fic centering on tweens or teens, and Kutie Pie was pretty dead on regarding the major differences. Having just turned 27, though, I can think of two major reasons age can make a major difference.
One is neurological and psychological differences among age groups. The differences are small and largely contextual (or "situational") between older tweens, teen characters, and even those in their early twenties, but there are two major changes that have a significant impact on character behavior and story arcs. One is cognitive and one is metacognitive.
Most people stop getting "smarter" around the age of 15-18. Their raw cognitive power, their ability to process things, plateaus at this point. There are cognitive improvements after this point, but they're generally not going to change your IQ score. A character who encounters a puzzle that they are intellectually unable to solve at 16 is unlikely to ever be "smarter" and thus triumph over the puzzle. A character who can solve that puzzle, but does so too slowly, would likely be able to solve the puzzle on time in five to ten years, as the young adult brain is in the process of making itself more efficient with the raw power it has. A character in her early to mid 20s deciding to come back when she's "smarter" is going to turn off any reader who understands this. More educated, sure. Better practiced, sure. Better at making the quick decisions? Absolutely. Smarter? Good luck with that, honey.
The other major plateau happens around 25, and pertains to impulse control. Most people have the best self-control of their lives in terms of risk-assessment and decision-making at 25. This is why your car insurance goes down at that age, and it's supported by studies. An extremely impulsive character at this age is going to be more likely to seem untrue or immature to an adult reader, unless we specifically see evidence that they were far, far more reactive when they were younger.
This isn't the main reason I think a journey fic with an older character is going to be much different, though. The biggest reason is cultural context, or what's going on in society around them. Past puberty, people show a remarkable ability to adapt to the roles that are given them regardless of age, but they generally fit themselves into their society. A thirteen-yea- old was often considered an adult in ancient societies, and nowadays some 20-somethings aren't really seen that way.
But if everyone your age is going out and going on a Pokemon journey, your story is going to feel much different from that of someone your age who is hopelessly long in the tooth to be on an adventure.
I'm writing something about a Late-20s starting trainer right now. Instead of being a story of hope and optimism, of a journey laid out before her, it's a story of redemption and triumph over shame. She's embarrassed to be what she is, and self-conscious about her age. She's ridiculed by other characters for starting out so late in life. Sometimes she looks at the kids she's passing who are just starting out and ponders the fact that she could have had children that age. In fact, I'm creating a subtext suggesting that she's using Pokemon training as a replacement for the children she probably can't have. That wouldn't really happen with a story of a ten-year-old trainer; it couldn't. And if you write a story where it's normal for 16- and 17-year-olds to go out training, it wouldn't happen there either. On the plus side, I'm having my character have a much easier time with some of the typical strategic struggles of a new trainer's journey because she's cognitively mature. The only intellectual struggles she has are of a very social, battle of the wits type nature, because she attained a decade ago an intelligence that's five years away for your typical ten-year-old trainer.
If you're thinking about the world around them, the age of a trainer is hugely important in any genre, especially journey fic. But if you just treat whatever age they are as default, then it matters much less.
I don't see how they can be better than ten-year-olds. You could have older characters who are just getting their own Pokémon because of various reasons. Unless they're exceptionally talented, they're going to be novices, and it will show in their actions. Even if they went to school or have been around them for a long time, I can still see these graduates/adults struggle at some point down the road in something--what that is depends on the character's weakness(es).
I think better here is an inappropriate comparison. Personally, I think older characters are more interesting just because they aren't done to death, but like you suggested, only if they're done with a certain amount of intention. I think the particular struggles of a novice trainer will be different depending on that trainer's age; older trainers will have a difficulty with plasticity or adaptability that younger trainers won't, but might breeze through puzzles and challenges that would stump a kid. But the emotional content of their journeys are going to be a lot different, and as an older reader, I would much rather experience the nuance of a well-written older character. Still, if the only reason your character is 20 is because you typed that number in the text, I'd rather read the kid, you know?
It's the setting that decides what the age for journeys will be, not viceversa.
Solovino, The rest of your quote more than adequately explained your point and provided enough wiggle room for someone who wants to write something interesting, but it made me think of something I wanted to point out. Where in most of the games does it say these kids are explicitly ten? Manuals, sure, but in every other game fandom to which I belong, manuals are treated with suspicion. In most of these games, the place where it says your character's age, if it exists at all, is subtle and hidden away. It allows a certain amount of wiggle room to write the story you want to write, the same wiggle room it allots to players of different ages who still need to relate to the silent hero they're playing. A journeying trainer in their early to mid teens does not strike me as inappropriate unless the setting makes a big deal of it, but a much older trainer established as having just set out does damage my suspension of disbelief. I'm sure it's the same with most of us.
Anyway, onto my particular responses to the original questions
How do you approach the age of protagonists' in fan fiction?
I generally try to avoid writing original character fic unless a canon character won't do for sheer exposure reasons, since many older fans refuse to read original trainer work. When I do need to write the canonical game protagonists, I try to use context clues from the setting, and place them on the higher end of what I find to be a believable range for my convenience. I generally treat the BW protagonists as the oldest, followed by XY, then BW2, then RSE, then DPPt, then HGSS and finally the first gen. Given that Cheren shares my profession in BW2, and that I felt hopelessly old by the time I was certified, I have a hard time imagining his crew as being anything less than 16 by the time the second Unova game happens, which means he must have been 14 during the first (and that still has him as something like seven years younger than I was when I certified. Go figure).
For original characters, it mainly depends on the story I want to tell. A journey story or a story of simple friendships and triumphs over evil will necessarily feature younger protagonists than a story about redemption. The political subtext and text I often incorporate often lends itself to older protagonists as well.
The main question revolved around journeys, but how does age affect other categories?
The main thing about character age is that it suggests where you are in life. It frames your hopes and dreams, and your shorter term aspirations. A ten year old is probably not thinking about starting her family, but a twenty year old might be, and a thirty year old almost certainly is, if she's not thinking about how she hasn't or never will. Some kinds of stories will only work with certain ages; others have a different flavor depending on the age of your protagonist. This is as true of Pokemon as it is any other property.
I also think political issues take on an interesting light when you use an older character, just because of the cognitive factors and life experience. Still, how many Pokemon fans write politics? Aside from me, I mean.
I think the rest of the questions I asked are more than covered in my responses. Apologies if my tone came off kind of rough. I'm used to arguing this stuff in a much more adversarial place, but this is the highest-level debate I've seen on this topic anywhere and I'm loving this level of discourse. You guys have amazing insights on this, and I am so glad I spent the day sitting here being intimidated by the forums just to find this gem!