About the Time Attack, the main speedrun strategy for the stages is exploiting the fact that you get an additional jump after performing a just attack with the bow in the air. There are many shortcuts that can be taken through this method, the biggest one is that you can completely skip the elevator with a crank mechanism in stage 4. Without any mistakes, it can be done in about 25 seconds, saves over a minute. I initially thought that the fastest horizontal movement in the game was the recoil from the diagonal aerial shots, but it turned out that walking is the fastest movement. The rock has the highest firepower of all the weapons, but its lack of shortcuts and poor ability for fighting trash mobs makes it awkward to use, so I only use it for stage 1, 2, and 8. Everywhere else uses the bow. Weapon route starts with bow, gets rock in stage 1, gets bow in stage 2, gets rock later in stage 2, gets bow in stage 3, stays bow until stage 8, getting the rock at the very beginning. I believe that grabbing the rock in stage 5 and switching back to bow in stage 6 can save time, but it's very marginal, and while the rock makes quick work of the catapults, it's all too common to take deaths there due to all the mobs. So I gave up on using it.
The rock does increased damage at close-range, so it's more of a melee weapon than a ranged weapon. It one-shots the rock mobs and the gargoyle girls, and most bosses are done in with just 6 hits when used like this. With the correct strategy and some luck, it's possible to skip the stage 2 boss' background phase. The stage 3 boss can be stun locked by the bow (the rock can do this too) with 100% consistency, you just to get the timing right. Stage 4 boss can also be stun locked with the bow, but in her case it's random, if she tries to charge then she might not flinch. She charges less frequently at close range, but there's never a 0% chance. I asked Nanakusado about it and he said the way the flinches worked was a little complex...
For the stage 6 boss, you can throw the plant bombs vertically. Two bomb plants + two just arrows from the bow are enough to start phase 2. For phase 2, it's just feeding her 3 bomb plants to trigger the fatal finish. She doesn't cause contact damage while swallowing a plant, so you can pick up another as she's swallowing it, and keep her on one side. This makes it easy to trigger the condition for the fatal finish very quickly. While she's blowing up, pick up a bomb plant in preparation for the gargoyle girls. By tossing two bomb plants around their landing points, you can defeat them instantly by detonating the bombs. This usually doesn't work because the timing is a little strict, but it'll still cause good damage to at least one of them, possibly even wiping one out. For the stage 7 bosses, Charlotte can be flinch locked in a corner with the bow if you get the spacing correct. Although the condition for her fatal finish will be met, it's faster to just ignore her until she dies. Direct hits with the arrows cause more damage than the explosions. By positioning yourself under where Hellhound shows up, you can get a just arrow before he lands. Hellhound will flinch after every direct hit with a just arrow. But if you miss the timing a little things get hectic real fast, that whip can flinch lock Youki to death, and it blocks the arrows so that they don't hit him directly. He won't flinch from just one explosion, it doesn't do enough damage.
The battle with Dran is the biggest luck factor in the run. Although the rock is the fastest weapon, Nanakusado actually designed Dran to counter it. She has a dedicated quick tail whip for deflecting the rock, so it'll be difficult to get a direct hit on her. When she breathes fire and when she lands from a jump are the best opportunities, but also when she's struck by the minotaur girl's attacks, and especially when she's grabbed. That's 2, sometimes 3, rocks, and Dran goes down in something like 6-7. Furthermore, the rock has friendly fire properties on the minotaur girl and Hellhound. For the final battle, switch to the ofuda talismans on the right side and guide Dran to stay on the right, then go to the left when she crouches. This way, she won't be able reach Youki with her grab attack if you go all the way to the left. Grab a bow from the blue box and fire arcing shots for a little bit of damage. Ideally, Hellhound's super will be delayed due to getting attacked by Dran, so that Youki can perform her super first. Hellhound's super charges faster than Youki's because it charges during the attack and during the conversation scenes in the fight. With the delay from Youki going first, they both get their super at roughly the same time. After that, it's hoping that Hellhound uses the next super (or the one after that) while Dran is crouching and the minotaur girl has the catapult ready. If you have the minotaur girl attack Dran while both supers are active, the damage is something like 60% of Dran's HP. It is essential to avoid getting grabbed when Hellhound has his super ready, because of the huge potential damage loss from not being able to use Youki's super at the same time.