I've had cats for 30+ years and none of them have ever scratched furniture and all have let me clip their nails; can you believe that? So, trust me, this works!
I like the using mayo to touch their feet suggestion. That was interesting.
I had one cat who hated his feet being touched. I got him when he was 2 so I thought there was no way I'd ever get to touch his nails. When he slept I'd stroke his paws, and then over time was able to stroke them when he was awake. I never solicited play with his paws. Soon - 6 months maybe? - I was able to cut his nails.
The key to using a towel with a cat is this: Do it quick, and do it tight. It's really best to do this with the aid of someone else, so they can hold the towel while you do the work. We call this the "kitty burrito."
Lay the towel on the table or floor and put the cat on the towel, including its hind legs and front legs. Wrap the towel tightly and lift it all at once. Keep the cat tight against the body (elbow holding rear to torso, arm holding body to torso, hand holding chest and head to torso). Pull the leg out that you want to cut. The other person should press the elbow of the cat to extend the leg; you take the paw and extend the claws. WHen done with that leg, push it back in the "burrito" and pull out another leg.
I only had to burrito my cats a few times before they let me do it either by me just holding them, or by sitting over them with them on the floor between my legs (feet together so they can't back out). The trick is to keep them in as natural a position as possible. If you put a cat on its back it'll want to right itself. If you pull the leg out to the side it will want to extend it forward.
If you're not totally confident and/or the cat is really not liking this, clip just the tips of the nails. Just a bit - just to give you confidence and let the cat see that this is nothing painful or threatening. THE MORE YOU CLIP THE LOWER THE QUICK WILL GO so each time you cut you'll be able to go a little lower. The shorter the nails, the better!
Scratching is essential to a cat's well-being and health. Scratching is their way of saying "mine," of saying "welcome," of cleaning and sharpening and dulling and pulling sheaths. You can't keep a cat from scratching, but you can redirect it.
SCRATCHING: THIS HAS NEVER FAILED FOR ME so I urge you to try it!
First, carry a squirt bottle at all times. When you see them going after something you don't want them to, say NO sharply (this works for anything). When they stop, praise them like crazy, then put them where it's OK for them to scratch. NOTE: if they're vertical scratchers, you have to put something up on a wall or a stand, and if they're horizontal, you have to put something on the floor. You can't change that part of their behavior.
If they keep going back to a spot to scratch or ignore the "no", then say "no" and squirt them. Try to be as far back as possible and not in their direct line of sight so they don't associate you w/the bottle. When they stop, put the bottle down, praise them, and put them where you want them to scratch. Keep doing it until they catch on. I've only had one cat who didn't get the hint after the first couple of squirts. Eventually they'll associate that action with getting wet. If you have a real stubborn cat who doesn't care about water, using 25-50% vinegar usually works.
SCRATCHERS: What does your cat go for? Soft textures or rough? High, low, horizontal?
I have an indoor/outdoor throw made of jute twine. My cats both started going for it as well as the carpet as soon as I brought them home. I managed to break them (in 2 weeks!) of doing the carpet and just decided to let them have the jute throw.
THere are scratchers you can get that are an 18-24" board w/ rope (sisal) wrapped around them with a rope hanger for a doorknob. My cats never liked them on doorknobs because they're unstable. I have one on the floor and one nailed to a support post. I watched my cat stretch and then hung it at the right height. Since they both also like upholstery, I put a piece of carpet remnant on the floor (NOT on the rug) as well as nailed a piece to the post (a corner of a wall is a good place to do this), too.
Then I bring the cat over to the board and put him on it, and make his paws like they were scratching, saying "good boy, good kitty" over and over until they get sick of me. Then I do it with my fingers, again saying good boy. Every time the cat walks by the scratchers I do this; first them, then me. Eventually just me. In a matter of DAYS all I have to do is say their name when they walk by, scratch the thing(s) myself, and they jump up and do it, too!
Within 4-5 days they will do it themselves without any encouragement. When I see them do it, I go nuts with the "good kitties".
BUT... there's still their desire to scratch the furniture and carpet...
OK. I've given them a place to scratch in the textures they prefer. I've reinforced their need to scratch and praise them for going where I want them to go.
Back to the squirt bottle and staying on high alert.
There's also a product - StickyPaws - that's like double-sided tape for furniture. Put it where you see them scratch. If it gets too expensive or there are too many areas, you can get large rolls of plastic wrap (if you have a Costco or Sam's Club they have industrial size boxes of really good stuff) and literally wrap the stuff around your chairs and couches.
The tape and/or wrap will not be there forever. They will soon turn their attention to the scratchers you have set up.
It's about staying consistent and staying on high alert. I'm driving my husband nuts with how I'm constantly creeping around after our new cats with the water bottle; he's never seen me train a cat before. But I brought 2 cats home 15 days ago, and the boy is addicted to the scratching post and the girl is still going between it and the furniture. But the point is, she's going between them!
Re: cat nail files/etc. - rope (jute, sisal) is great for dulling, where carpeting is great for pulling sheaths and cleaning the nails. There are cardboard scratchers that are kind of in between those textures. I've had cats who didn't care for them and others who did.
Remember: you can't stop a cat from scratching. Where and how much, though, you can control. Keep their claws as short as possible. Constantly encourage them to use the scratchers. And praise.
Cats know they're wonderful; they need to know that YOU know they're wonderful, too!
Please let me know if this works for you!