Did a playthrough without using non-mandatory badges and power-ups besides the normal mushroom. I couldn't find a way to get these collectables:

• W1 - Pipe-Rock Plateau Palace, 2nd 10-flower coin: Requires Elephant to break the blocks around the coin. There aren't any Koopa shells area to break them. [1]
• W2 - Cruising with Linking Lifts, 2nd 10-flower coin: Requires Elephant to break the blocks containing the lift blocks. [1]
• W3 - The Anglefish Trial: Ready, Aim, Fly!, 3rd 10-flower coin: Requires Elephant to grab the large shell and break otherwise unbreakable blocks to access a pipe. [1]
• W4 - Sunbaked Desert Palace, 3rd 10-flower coin: Requires Drill to reach the coin. Possible to reach in theory, but the sand lowers your jump height. [1]
• W4 - Break Time! Treasure Vault, Wonder Seed: Requires Drill to access the level. [1]
• W6 - Where the Rrrumbas Rule, 2nd 10-flower coin: Requires Drill to access an offscreen area. [1] [2]
• W6 - Where the Rrrumbas Rule, Wonder Seed and secret exit: Requires Drill to access. [1]
• W6 - Raarghs in the Ruins, 3rd 10-flower coin: Requires Drill to reach. Might be possible to reach the area without Drill, but you still require it to reach the coin. [1] [2]
• W6 - Badge Challenge Boosting Spin Jump II, all collectables: Inaccessible because of the above.
• W6 - Expert Badge Challenge Jet Run II, all collectables: Inaccessible from the lack of Wonder Seeds needed to break the gate.
• W6 - Expert Badge Challenge Invisibility II, all collectables: Inaccessible because of the above.
• W6 - Expert Badge Challenge Spring Feet II, all collectables: Inaccessible because of the above.
• W6 - Search Party Item Park, Wonder Seed: Requires the use of power-ups. [1]
• W6 - Break Time! Hot-Hot Rocks, Wonder Seed: Requires Elephant to break the entrance blocks. [1]
• Petal Isles - Blewbird Roost, 3rd 10-flower coin: Requires Bubble to touch a flower that contains the coin. [1]
• Special World - Deep Magma Bog Special Solar Roller, all collectables: Inaccessible because of the above.
• Special World - The Final Test Wonder Gauntlet, all collectables: Inaccessible because of the above.
• Special World - The Final-Final Test Badge Marathon, all collectables: Inaccessible because of the above.

Made this list if anyone wants to speedrun without badges or power-ups. It's worth noting no collectable requires a badge that isn't forced to be used.

Second coin in downpour uproar can be attained by carrying a shell

Verified that, thanks, removed from list.