Firstly, the general story idea is fairly interesting, so that's a good start.
Now... I will respond to this first:
A third character will probably be introduced in the next story (assuming I finish this one) to make a trio - and I'm sure some romances and stuff will come into it at some point. The stories themselves will probably feature more conventional violence and less pokemon battling than the games and anime, though there will obviously be some battling as well; after all, Maylee is going to become a Pokemon trainer and will end up in gyms etc when not saving the world.
On the first point; how necessary is this? The way that's worded it seems that you're thinking about a third main character just to make a trio, and maybe introduce a romance, etc into it, but I'd warn against doing it just for the sake of that. Be sure that if you add a third main character that they would be relevant to the story and add their own impact, rather than just add a romantic sideplot. Also note that in fact the trio set-up with Pokemon fics with some romance, etc is
very common. This doesn't mean you shouldn't do it, but it does mean that if you do it then if it feels just like a repeat of how those other fics do it readers may get turned off from it.
For the second point, just be sure that it makes sense for Maylee to want to take on gym leaders and has time to do that and it makes sense/is realistic for her to do so in-between 'saving the world' - don't try to rely too much on following the formula of the games after all.
Secondly if any, more experienced, writers have any ideas for scenes or methods of writing to make Maylee's personality come across better to the reader - any tips are also appreciated.
Well, that depends on how exactly you want the reader to see her or rather what you mean as 'come across better'. =p On that note; don't try to overdo it. If you have this character be protrayed as far better than anybody else despite her flaws then the character will actually suffer for it and seem boring. Nobody really wants to read about the person who does well at everything, and sometimes a likeable protaogist is still one which has their flaws show.
Thirdly, I worry that I don't have a way of making sure that Maylee is very obviously the main character despite her appearing first
I wouldn't worry too much about that, as long as you don't focus too long on Jack beyond the prologue. You could also say try to soften the amount he is involved in the prologue; for instance, keep description of him vague, don't yet name him, etc, and leave that for later after you establish Maylee's character.
Fourthly, if someone could perhaps help me pick a good pokemon for Jake - I was going to go with Pikachu but since that's really common I thought I'd give Marill a try, but Marill's a little bit odd. I'm not too familiar myself with the newer generation pokemon so perhaps one of those is Pikachu-like (small and feisty-looking, and preferably not normal or psychic or ghost type).
There's not too many right or wrong Pokemon, frankly. I would suggest however against Pokemon that are very rare or unlikely for a regular trainer to start with (say a Dragon type or Eevee) without a solid explanation, and that'd extend to legendaries too. Marill as a side note is not too odd a Pokemon to read about; if a canon example of HGSS's pseudo rival having a Marill exists then I doubt there should be many problems with it, and it's also not a rare Pokemon for many regions, etc.
There's a bunch of Pikachu-like Pokemon as well that'd fit that bill; Plusle/Minun spring to mind as does Pachirisu and Emolga from 5th gen in particular. As for non-electric types... well, the list is long! Try some research into it and see what appeals to you to write about; there's the likes of serebii, bulbapedia, etc to use for that.