ceiling first - along with any ceiling light fixtures

Then demo cabinets, install new floor, install new cabinets

If you have mold issues, you probably can't "clean" it - the drywall or plaster behind the existing cabinets will probably need to be replaced. If there was ever a plumbing leak in that area that went unfixed for any length of time you most likely have mold. Best to bite the bullet and eliminate it.

You will like having new cabinets. I'd caution you about Ikea stuff though - it looks great but that stuff just does not hold up, and in a few years you'll wish you got real wood.

Be sure to take lots of before and after pictures. We all want to see.

Here's a trick - if you can't see into the cabinet to see if there is mold, stick your camera in there and snap some pictures - if it's there, the mold may show up in the photos. Or it may not - but costs nothing to try it.