I put on an old sock that I have dipped in soapy water and run my hand down the strings to clean them.
To repaint knobs, remove from furniture, clean well, rough with fine steel wool, then spray paint. I usually put the screw back in them and tie a long piece of yarn so there is a loop to hang the knob. Then I suspend from a dowel and spray paint (photo). This allows me to get paint on every side of the knob at once for an even finish.
Borax usually. Maybe oxi clean? I wonder if a magic eraser would work if you took down and lay flat, like on a counter top. Once you figure it out, post it. Also use paint for plastic.
To repaint knobs, I've had good success with having the screws in the back of the knobs and poking it into a sheet of styrofoam to hold them up. Then spray paint them.
If you can take the blinds right off, summer is the best time, I spray mine outside with simple green or spray nine, let them sit for about 2 minutes then use garden hone on watering spray to rinse them. Make sure you spray the strings when you are spraying on the cleaner, let them drip dry outside then rehang. You can also try to do this in your tub but it will take longer - good luck!
That is too bad, but I think there is hope, you may even want to try to dye the strings a light gray, and try to put a coat of clear nail polish on the metal pieces to prevent them from rusting on the string
What kind of knobs? You can likely just use an acrylic paint for them
Wood. Some are broken. I would replace, but found they are like $5.00 a piece. That's ridiculous! I really wish there was an alternative to these blinds. Perhaps move to the middle of the country in the woods where they are not needed.