I would like to buy this (or any other 3/8 compression valve, not to make this specific for this product) https://www.lowes.ca/product/38-in-dia-chrome-compression-quarter-turn-straight-valve-586694 Does it come with a compression sleeve and with the tube insert shown in the images below. Also does it work with both copper and the below 3/8" pressure rated tube?
A worker at the tool store can tell you for sure, but generally compression fittings include everything you need to complete the connection, assuming you choose the correct fitting. The tube insert appears to be a brass tube sticking out of the compression nut in the picture. Your diagram shows the parts separately because it is a guide designed to aid in a minimal repair.
Assuming you are talking about using this valve for the application in your picture where it shows a valve, it will not work.
It is the wrong valve for your application. It is 3/8" compression on both sides.
The valve you are looking for is much more common than the valve in your link. You need a 1/2" nominal comp. inlet (5/8" o.d.) x 3/8" comp. outlet, 1/4 turn, straight configuration. Any decent plumbing shop will have what you need, although I can't speak for low-skill/high-volume "bix box" hardware stores...