I think it’s fairly well established that use of the Force is not a game of top trumps.
Conjecture like “if Rey is awakened why couldn’t she stop the force push on to the tree” this isn’t a video game where you have force levels and you automatically stop Force Push attacks. She’s not really expecting Kylo Ren to suddenly throw her against a tree and is not making any concerted effort to combat that.
Rey beats Ren because she finds some strength in the force, who knows maybe Qui Gon Jin is talking to her or Luke is doing something it’s not really important, she takes him by surprise and as he’s not in a focussed state of mind and is already pretty beat up.
“She’s not ready” is not the theme of THIS film. If anyone is not ready it’s Kylo Ren, masterful with tricks and wizardry but prone to mistake and rash behaviour. Instead of finishing her off when he had the chance he gives her time to think, in hope that she can become an apprentice.
As for the “lightsaber belongs to me” it’s a family heirloom to him, I got the impression he knew who’s it was - I guess there could be a bit more to it than that.