I don't have these problems but here are some thoughts on why I don't basically based on a few years of making a mess
1. I don't leave the meat to sit in one spot for more than 2 hours or so. IF you do, and it sits in one spot for the whole cook it will stick and more likely will tend to soften and mold into the grates themselves. More the meat occasionally.
2. Fat side down tends to make a mess if it's at the end of the smoke.
3. Overcooking means over tender and will more likely fall apart.
4. Don't use tongs on big cuts of meat. IF it's too tender tongs tend to rip more than anything.
5. IF you use foil (which I don't much) the likelihood of a hole appearing is directly proportional to how long it's been in the foil and how large the cut of meat.
I use a pair of cotton gloves on top of which I use a pair of rubber gloves. I have a table right next to the smokers and I pick up the food and place it on or in a pan for transport.
If I know it's going to be messy I move the table closer.
For my birthday, my wife bought me one of those plastic food service carts (from restaurant supply) and I use that to take a lot of food back and forth.
Truth be told the spot right in front of my smokers does have some grease stains.