There is never a reason to enter forest0_door1. At most, it allows to you see what item is in comp2, but you cannot collect it. The only way to collect that item is by going all the way around the loop starting with shed0.
At lake0_door0, there is no point in going to the right if you do not have the Teleport Wand. That way leads only to shed0, which requires the Teleport Wand to make any progress.
If you don't have any Blastola Colas when you enter lake2, wait at the lower level until the enemies spawn down there. That prevents them from all spawning at the top of the screen where there is no room to avoid them.
If you have even one Blastola Cola, it's pretty easy to shoot all the enemies even if they spawn at the top. If you have the Boots or Teleport Wand, you can take the lower path as well.
You can get over the white brick hill by the rocket in lake2 using either the Boots or the Teleport Wand.
The randomizer logic will never require you to go through the castle without the Lantern. But if you find the Door Key right away in forest0, you may as well take a peek inside the castle. Even though the level is all dark, you will be able to see what item is in castle0.
If the game gives you both the Door Key and the Lantern, and you haven't gotten the Boots or the Teleport Wand yet, it means that either the Boots or the Teleport Wand will be at castle2_item or castle0_item. The reason for this is that it's impossible to get out of castle2 without either the Boots or the Teleport Wand—the gap at the left is too large to jump without the Boots, and the door on the right leads to the castle0 item room that can only be escaped without the Teleport Wand. The randomizer logic knows about this softlock situation and ensures it can never happen by ensuring that if you can get into castle2, you will have the tools to get out of it.