In this case, may I request a tiny bit of advice concerning Yumemi? Anything would be helpful, really. You see, I decided to try PoDD Normal today (my first or second time) and went no-miss through stages 1-8, then Yumemi murdered me mercilessly. Is there any set amount of time you need to survive before hoping to defeat her, anyway? And does the fight get any easier with further attempts?
Alright. First of all, the way things work on Normal (maybe I should have mentioned this before) is that the stages start off very slowly, and as time goes on everything gets faster and the enemies get healthier. Most of the time, the AI doesn't live long enough for the game to reach top speed. Yumemi of course lasts an extremely long time, and the speed doesn't completely reset after you've lost to her, so the game itself actually plays a lot differently against her than it does against the earlier opponents. Since her own attacks aren't overly dangerous, I suspect that the difference in speed is likely what causes people to get destroyed by her on Normal, while they find the rest of the game comparably easy. So I think what's most important is getting accustomed to the higher speed and healthier enemies. You'll need much different timing when destroying enemies to reflect fireballs, mainly because there's a significant difference in the way the enemies chain themselves. In the early game they work kind of like PoFV fairies, and the explosion from the first one quickly burns through the entire formation. Against Yumemi they start acting more like they do in PoDD's Lunatic mode, where the explosion effect stays in one place as the enemies quickly rush in to "renew" it. It's also much easier to get suddenly walled by fireballs, since they'll be moving so much faster. I'm not really sure what the best way to practice dealing with that is, but playing a bit on Hard/Lunatic couldn't hurt, I guess. To be honest, when I play Normal, even with my countless hours of practice with Lunatic, I still think the shift in speed that happens against Yumemi is pretty jarring.
Anyway, about Yumemi herself. Her spell attack is aimed at you, and if you're moving around as it's fired, it'll be a nightmare to get through it. Whenever she uses a spell attack, you should stop moving, get a charge attack ready, and just sidestep after the bullets are fired. I can't really stress enough how important it is to use a charge attack in these situations where you're forced to stop moving.
Always use a charge attack as you go through Yumemi's spell attacks, because there
will be fireballs coming straight down at you. Also, while you shouldn't move horizontally as the bullets are being fired, it's okay to move up a down a bit if you have to, since that won't really mess up the spell's aim very much.
Her boss attack is pretty annoying to deal with. She attacks slowly, but she's constantly moving, she's very healthy, and she barely shoots any pellets, so you can't reflect her away. She has five attacks, and I guess I'll try to describe each of them. First, there's one that shoots star bullets in a spiral-ish pattern. (Shinki stole it in Mystic Square.) While it's totally static with two possible orientations, clockwise or counter-clockwise, you probably don't need to know that for Normal. It's one of her easiest attacks since the walls don't connect like they do on Lunatic. Then there's another where she shoots stars off the top of the screen and they just fall randomly. There's nothing predictable about it. (Something that's kind of interesting is that if you destroy her as she's shooting the bullets up, they won't fall back down after she's gone.) Both of these have no reflectable elements at all, so you're really just forced to evade them.
Now these next two look similar at first, but it's very important that you be able to tell which is which as soon as she starts using them. Both start with her aiming a thin laser beam at you. First, the targeted cross attack. The delay before she uses this is higher than it is for any other attack, so she'll seem to keep floating around for a bit longer than usual. Then she stops moving and aims the laser at you, and you should immediately notice that the laser follows you when you move around. There are a number of ways you can misdirect the crosses, and the best method will depend on how much room you have. If you're somewhere in the middle of the screen (and probably will be), then try to move midway up the screen, then across to one side, and from there you can kind of "stream" the crosses back across the screen. (Again, it's just generally good to keep moving left and right, and only stay on one side for as long as you have to.) If you don't move up and instead just try to rush to the side, she might start spawning the crosses as you go, and they'll block the path behind you, trapping you in the corner. If you moved up before you rushed across then you can easily back up and pass underneath any crosses that started spawning. Then as you stream them across the bottom you can move in a few quick lurches, since the crosses don't materialize very quickly. Don't just run charge straight to other side or they might trap you if they aren't done spawning. So if you have enough room, that's probably how you should deal with that attack. If your space is really limited though, you can try to stream the crosses diagonally. With just some very tiny taps you'll keep ahead of them, since the arms won't be hitting you. Try to start this from as high up as possible (always easy to back up) and stream downward, rather than starting at the bottom and gradually moving up. Unfortunately, if your movement was limited to begin with, you'll probably end up starting this at the bottom of the screen. It's unpleasant, and leaves you very vulnerable, so try to avoid having to dodge them that way.
Now the other attack with the targeting laser can be identified by seeing that the laser doesn't follow you after being initially aimed. For this one you just want to get the hell out of the way, since she fires a big beam of bullets in that direction. She actually stops moving for a bit when she does this, and some of the bullets she fires are reflectable, so this is probably the attack that leaves her the most open to charge attacks or being reflected. So this looks similar to the cross attack at first but be sure you identify which one it is before she fires.
And the last attack is fairly ridiculous. Just a huge spam of random crap. (On Lunatic, the game usually hits the bullet limit whenever she does this.) She's even constantly moving when she does it so it's almost impossible to reflect it away, since she'll just move out of the explosion. I'd treat this like Chiyuri's thing and just bomb it always. Or maybe it's easier on Normal. I think the bullets might be extremely slow or something, but then you still have the issue of barely being able to move. Well, bomb if it's going to hit you, I guess.

And now really. I was expecting this post to be much shorter than the other one, but I rambled again.
Oh, and about your other questions. I'm not really sure how long you have to last to beat her, and it depends a lot on your character. I'm pretty sure the time gets shorter as you lose to her, though the bullets won't get slower like they do in PoFV. And don't expect to win the first round.
I'm having trouble with capturing Hourai Doll, despite best efforts. Please help this poor soul.
Move in a clockwise circle to avoid the aimed bullets. Beyond that I'm not sure what to say about it.