I believe oil on the latch and shelf allow motion, even to the point of having the action open up when the shot goes off. Friction is what holds the latch and latch plate together, and oil defeats that. I had a NEF .30-06 refinished with TR Gunkote which is a Teflon finish. When I got it back it would pop open with every shot. That didn't stop until the gunsmith cleaned all of the finish off the latch and shelf after which it was good to go again.
I'm not familiar with the scope although the little ground squirrel rings a bell. Is the 10 - 300 and infinity perhaps a parallax or focus adjustment? I would bet that it is parallax, since you would have noticed visual changes if it were focus.
I assume that you have checked that all the scope mounting screws are tight. You will need to try different brands of ammunition if you are going with factory loads. Also different bullet weights. If you are using reloads, some of these rifles are very load sensitive. Again from personal experience, I load from 20% below maximum for a given powder, five rounds, rubber band them together with a piece of paper with the charge weight, then increase the charge by one grain and load another 5 rounds, bundle them together with their charge weight, and keep this up to maximum load. This assumes you are loading the same bullet, you only want to change one thing at a time. Then go shoot each bundle of five from a bench, allowing the barrel to cool between them. It is classic to watch the groups shrink until you reach the load the rifle prefers, then start to open up again. My first H&R was a .30-30 model 158 and it went from 6" groups down to almost 1" at its preferred load with 150 grain spitzers.
Good luck, and enjoy playing with the little rifle! Let us know how you make out.
-WH-