Not fully backwards compatible, their is a list of games it will play from 360's library but it won't play all of them. Xbox adds a few new backwards compat games each month though and they have added many Orignal Xbox games as well.
In short, Xbox One S will play all of the Xbox One library.
And here is what it will play outside of that. (360,Xbox OG)
https://en.wikipedia.org/wiki/List_of_backward_compatible_games_for_Xbox_OneAs for your concerns with 360, I have never heard of a large scale issue with the Slim models. I know the original models were notoriously one of the least durable and poorly constructed consoles ever made with major red ring of death flaws.
Xbox One is an amazing console. It's worth the upgrade and than some. It has one of the best controllers ever fathomed and an amazing lineup of exclusives. Especially for racing and FPS lovers.
But with that said, XB360 Slim should last more than 3 months with casual use. And if it does break it's a cheap fix now adays. Ive seen shops willing to fix RROD for 20 bucks.