Wow, a very tough one to answer. I think of our dear next-door neighbors whom we lost and were sad to see leave; it was wonderful to have neighbors with whom we could leave a spare key when we traveled and who would pick up our mail or those pesky "free" newspapers that litter the driveway if we were away. Those are the kinds of things that are both helpful and cheering to know your neighbors will do, and you don't want to burn bridges with what sounds like a nice, helpful but clueless family. (But..our neighbors never would have wandered into our yards!)
Someone mentioned putting away your outdoor furniture but that may be a hassle. I would get inexpensive weatherproof covers for the furniture and keep it covered; when you want to use it, you can just whisk off the covers and replace when you're done. Some of these covers include cloth ties you use to tie the covers around the furniture's legs (others have snaps to secure the covers to thelegs); I would hope that your friendly but boundary-free neighbors might at least balk at having to stop, stoop, and actually untie tied-on covers before they sit on your chairs! If you do this and look out to see that they've uncovered the chairs, then yes, I'd go out and say nicely, "Hey, Bob, I know those are comfy but we've started keeping them covered so they'll last longer. When we have a barbecue we'll let you know for sure, but meanwhile, we'd like them to stay covered." I would hope that would make anyone pop up and apologize.
I do agree that if it's possible you could finish the other two sides of the fence. That may or may not work with the line of trees, though. I would at least consider it and get some estimates. I would not lie and tell them it's because you're considering getting a dog; just say that you'd like to complete the fence.
If you see them picking flowers etc. -- that I would stop immediately by saying cheerily, "Hey, Bob, please don't pick any flowers. You'll be able to enjoy them a lot longer if they stay in the ground!"
If they're as nice as you think, they will respond better to a cheerful correction than to a grave talk. I know you said you have tried being lighthearted and it hasn't sunk in. I would try again but be sure you're saying "don't" clearly in any statement you make, such as "Please don't remove the covers to sit on them -- I promise you'll get to use them next time we cook out" and then..do have a cookout and invite them sometime soon so they know you do like them!