I finally succeeded at killing the Mimic Queen with Vaan at level 12, Fran and Balflear both at level 13 and Basch at level 8.
Before that, I had been getting tougher and tougher, until it was fairly easy to make it down to the main canyon section. There, I killed a Battery Mimic, which put my battery up to 40%. After that, it didn't go down, and many of the tougher creatures just disappeared. So there was nothing left to do but fight the Boss.
Even though I'd hit a pretty high level, it was tough. Considering that Basch doesn't get experience points since he's a guest character, I'm thinking that there must be a trick to this, possibly involving water.
I'd hardly played using magic at all, so it really didn't occur to me until I realised that the one thing that really combats electricity well in Final Fantasy is water. But I didn't have water, so I used fire. It worked quite well. I customised everyone's gunbits so that they attacked whatever the leader was attacking, but even then, I was pretty much controlling them all throughout the battle. I kept using fire on the main creature and ignored the mini Mimics. Whenever anyone got close to death, I gave them a potion. And, when you're told that the Queen is going to use Thunder, if you dash down one of the longer passages, I think you can avoid the worst of it. Finally, I used a number of licence points, including one to activate a square that increases your HP by 50 points. It's worth noting that some of the best squares so far can be found on the licence grid by buying gunbit squares, otherwise a complete waste of money.
After that, everyone steps out into sunlight and finds themselves surrounded by sand dunes. There are plenty of monsters out there, but they're really easy now! Makes me think I might've overlevelled... So I presume there was a trick to that Mimic Queen and that I just didn't find it. It's worth noticing that my battery did start to drain once I'd started the fight.
On the way back, you can go to two areas labelled '????'. The first one on your route leads to a market with a fantastic cut scene. I reset soon after finding this one -- I bought lots of potions from the first shop I found 'just in case'. However, when I went into the main market section, there was every type of shop available. The other pasaage to '????' leads to Giiza Plains.
Anyway, you head back to civilisation and Fran and Balflear leave your party! I later found them in Tomaji's bar, but they weren't going to leave until my character found Penelo.
That's your aim -- to try to find Miguelo and Penelo. Neither of them are in Miguelo's shop, which is said to be unusual. You're also told that Miguelo didn't look well. You have to go to Daran's shop, which is right at the bottom of the downtown map. He's the old man with the cat-rabbit that you spoke to previously.
Along the way, I had a look at the sewer again. There's an area with a save crystal, surround by a number of posts. There are two lit symbols on the ground and one post is also lit. I wonder what happens when all the posts are lit...? I also went through a number of exits in that room, but they didn't lead anywhere -- it just seemed like a maze.
Daran gives you the sword that belonged to the man that married Ashe in the very beginning. I've forgotten his name, but you saw him being given it in a ceremony. I think I even took a screenshot of it at the time...
Anyway, I've no clue what to do now. Penelo could be pretty much anywhere. I have found an Airship Terminal near the West Gate though (Cid?). I ventured out in all different directions, but with no idea of whether I'm meant to be there or not, I'm really lost. Since I have a big sword, I guess I'm meant to fight something. I did look at the clan board in Tomaji's, and I have the option of fighting a wolf, but I've no idea where it is either... Maybe I should try talking to more people in the bar?