Same as Jorito on point 1, although I won't just take the lead melody, but other elements in isolation. A cool bassline can be a lead melody. Or vice versa.
I'll then iterate on whatever elements I want to use, over original backing, or in isolation. It helps to be able to play an instrument here, even if you're not very good at it, as it's much faster to iterate on the parts you're using that way. Just improvise, jam, mess around, and you'll come up with a lot of cool ideas that you'll want to incorporate into an arrangement. This is why my arrangements end up a bit loooong.
Then it's just a matter of tying them together in a way that makes sense. Sometimes I do that well. Sometimes... not.
Making "originals with stolen melodies" (great phrase) is a good way to break away from the structure of the sources. You can start from a rhythm not found in the original and add elements from the original to it, and adapt them as needed, or write new material to support them. A new rhythm goes a long way to preventing a remix form being too much like the original, even if chords and melodies are the same.