The big spread and the sheet are in the wash right now with soap and a cup of bleach.
I'm planning to do more or less what your mom does as far as sewing. I don't have a serger (overcast) machine and I don't have (or want) pinking shears, so I am just going to sew a double row around when I attach the back to the top, and I think that will be good as it isn't visible and just has to hold up for many, many washings.
EDIT TO ADD:
11 PM. Got one made. Looks like a playpen sheet. Can't try it right now because she's already in bed. Will wash it tomorrow morning and try it on the mattress and if it's OK will take a picture. Took maybe an hour and a half.
EDIT TO ADD:
8 PM Another one done, that's 2.
Here is a picture of one of her old sheets. You can see what happens when the fabric gets thin and she digs it. The back is still intact but the top is torn. She will dig-dig-dig-dig-dig. If I tell her not to, she stops and then goes back to it. She'll have a comforter and maybe another blanket in her playpen, but she'll get in the middle of the playpen and dig the sheet. It's got to be an instinct, trying to dig a hollow at night. I have another dog who digs her fleece bed, too. But at least I can get another fleece bed, unlike the playpen sheets. The groomer uses the dremel on both of them so their nails aren't sharp.
I don't know why I even saved that sheet, it was already retired, but it was in the bottom of her linen basket because I just knew I couldn't buy any more. I won't have to be scared to get rid of it now.
I don't know how long it will take her to dig through these new ones, it will depend on the fabric. There's nothing wrong with the quality of the Cozy Kids. They are already worn thin before she manages to get through the top. If I'm using 2-3 sheets at a time for her, that's 120-180 washings a year, so they just wear out. This is one of the reasons I don't use bleach every time. I'd like to but they'd wear out 10x faster.
I don't really sew that much, but I'll be careful if I do any fleece!
At this point the thing that slows it down is the step where you pin the back to the front to fold the corners. I have to do it on the floor because there isn't a table in the house big enough, and mostly my creaky old self doesn't feel like crawling around on the floor with pins, so then I put it off. But I got 2 pinned this afternoon so those can be finished now. Then there is just one more to do from this batch.
If I was smart, I'd keep going and make the other 8 (out of the other quilt/bedspread) while I have it all memorized how it's done. Haven't cut those out yet. If we use 3/year (I don't really know how fast these new ones will wear out) then making all 14 would be another 4-almost-5 years. Might not have to make any more till, er, 2024? I think I should do it. We'll see...
Posting the progress is kind of helpful even if it's super boring to read but people with an incontinent pet who sleeps in a playpen will probably understand.
EDIT TO ADD: 11PM fourth sheet finished