My new girl, Millie, is the same. She doesn’t like being picked up and she’s fussy the whole time. She whines and wheeks like I’m hurting her and she hates having a blanket around her. I suspect someone threw a blanket or towel over her because she gets nippy, so she’s on the defensive and bites the blanket. She gets so stressed she just gobbles the food down while she whines.
The only way to get them accustomed to being picked up is to......pick them up frequently. Get to know them and their preferred nap times for one. My Rafi is the ultimate lap piggy in the evening after he has had a substantial dinner, but he doesn't like being taken out of the cage during the middle of the day when he sleeps for an extended period of time. Cavies are as individual as we are!
With regard to clipping nails, I roll him snugly in a "burrito wrap" with just the target nails sticking out. Cut the back nails first as those are the most difficult to manage. Hold him firmly against your abdomen with your forearm and use your hands to manage the nail. He will probably kick and whine the entire time! This is tough love! They seem more tolerant to have the front nails cut despite them being close to the face. No matter how tame the piggy is, he may bite the hand that is holding his nail so I would wear gloves. After the process is complete, allow some time to hold and soothe him and perhaps have a tasty treat at hand.