First of all, I am very glad that you ended up getting two prairie dogs; I have read that they are such social little creatures that being an "only child" can be devastating to them. I, too, am a first-timer, having recently (almost three weeks ago) gotten three of them myself (one boy, one girl, and another--probably--girl).
It seems like my little ones don't mind being handled, but like yours, they squirm---a LOT. I have found that the best way to have them let me pick them up for a bit is to hold them against my body while petting their head continuously. While this doesn't always work, they often can't resist it and will be still for a little bit while they soak up the petting. This also helps with the scratching (their nails are sharp, aren't they?!). I also noticed that they squirm a lot more when you try to put them on their back (cradle them) than if you leave them face down (with their belly against your body). Also, this may be coincidence, but one of my prairies is about two weeks older than the other two, and she is easier to hold. I have heard that they mellow with age, so we'll see.
Speaking of sharp nails, I have not clipped their nails because I am so worried about cutting the quick of their nails. Iíll just deal with the sharp nails until I can think of a better way, or until they calm down more. If you wear a long-sleeved shirt, you donít notice the nails nearly as much.
Mine also climb on everything and when I have them out of their cage, I am constantly keeping them from climbing up the outside of it. Inside their cage, the maximum height that they can fall from is about 24 inches (a little less when you factor in the height of the prairie dog). It makes me super nervous, but so far they seem to be okay with falling from this height. I do have cardboard lining the bottom of their cage, which helps to absorb some of the impact. Maybe you can do the same in the areas where you cannot put a pillow.
With respect to chew toys, I was also concerned with splinters. My suggestion would be to make sure that your chew toys/sticks are made of hard wood, rather than soft woods that are more likely to splinter. I bought my prairie dogs two hardwood chew toys but to be honest, they show little interest in them. They would much rather chew up the cardboard that I use to line their cage. I do give them a lot of hay, though, so I am hoping that this will help wear their teeth down.
When I got my prairie dogs, they had been fed a diet of pellets (specifically the ones made by ExoticNutrition) and hay. I have been thinking about adding a little bit of high-quality dog food to their diet, but I hesitate to do so because I hear mixed things about the dog food. In fact, I am also a little bit wary of the pellets---the ingredient list on the bag is soooo long that it makes me wonder. Since Iíve had them, I feed them fewer pellets, more hay, and supplement their diet with organic vegetables (sweet potato and carrot). I have also been providing them with sunflower seeds for extra protein (rather than the dog food). They seem to be doing very well on what I am feeding them. I would suggest cutting back on the amount of dog food your prairie dogs are eating, just to be on the safe side. Have you tried feeding them anything else yet?
Hope this helps, and if I learn anything else useful from my own experiences, Iíll be sure to post.