Turn a signature on paper into a transparent PNG

The threshold is too low, so faint variation in the paper is being counted as ink. Raise it until the background disappears. If raising it also starts eating the thin parts of your strokes, the underlying problem is usually contrast - a photograph of a pencil signature on cream paper has very little difference between ink and paper, and a darker pen on whiter paper is the real fix.

Because a photograph of white paper is not white. It is a gradient, and any single cut-off either keeps the shadowed corner as a grey block or loses the strokes that fall in it - in a photo taken in normal light, ink in the shadow can be brighter in absolute terms than paper in the light. This compares each pixel with the paper immediately around it, so both ends of the page are treated correctly.

That depends entirely on the document, the jurisdiction and who is accepting it, and this tool is not the thing to ask. What it produces is an image of your signature with the background removed - the same thing you would get from a scanner and an image editor. Whether an image of a signature is acceptable for a given purpose is a question for whoever is asking for it.

Yes. Recolouring replaces the ink's own colour while keeping the shape and the soft edges, which is useful when you photographed a blue pen and the document expects black. The transparency is unchanged, so the strokes still blend properly over whatever they are placed on.

No, and this is the page where that matters most. Your signature is decoded, processed and re-encoded by your own browser on your own device. There is no request that carries it, and a content security policy the browser enforces prevents this page from sending anything to another domain. Nothing about it is stored, logged or transmitted.