You are using safari, right? (or another IOS browser)
Safari doesn't support Canvas filters.
I might one day implement them in plain JavaScript, but currently I'm not spending too much time on all the weird and crippling quirks of Safari.
Please use a modern browser like Chrome, Firefox or Edge, or at least one that doesn't actively tries to hold back the web.

Yes very often you do...
This is mainly an issue with Photoshop on Windows, which doesn't save the transparency information in the clipboard in a standard way.
Weirdly enough, Photoshop on Mac does preserve transparency in copy/paste.
As a workaround, you can first save your file as PNG and import that instead.

I mean: the toolbox is even on the wrong side!
Yep. It's certainly not a remake of Deluxe Paint, but it's heavily inspired by it, regarding Amiga spirit and pixel handling.
I like to think of as "What if Deluxe Paint was made today?"
Besides: Deluxe Paint on Atari has the tools at the bottom, blasphemy!
If you're looking for a more faithful recreation of Deluxe Paint, check out PyDPainter.

Yes.
Download the Lospec Palette as PNG, open that image in DPaint.js.
Next to that, you can extract the palette from any image by selecting the menu Palette -> From image.

Don't.
The "New Icon" format was a hacky and messy way to solve a problem 30 years ago, but nowadays there are far better solutions.
Save your icons as "Color Icon" (or "Glow Icon") and use a proper icon library on your Amiga.
Please stop using the "New Icon" format, the sooner it fades out, the better.
DPaint.js can read New Icons, but write support is not planned.

Apologies for being opinionated and pedantic.
DPaint.js is targeted at pixel art and lo-spec images.
While the JPEG format is fine for high resolution photos, it's not a good match if you want to preserve each pixel of an image because it's a lossy compression that blurs and alters the image.
Therefore, DPaint.js doesn't provide an option to save as JPEG.