The reason for all the lag you've experienced, is due to Intel's integrated HD4000 graphics being sub-par for gaming. Truth is, its actually good enough for some simple games believe it or not, so its not terrible. But for other games where higher FPS counts matter, it is not.
The reason you are not able to load the game on Steam however, is more then likely because you've neglected to engage Steamplay compatibility and force a Proton version. Here's how you do it. Go to your Steam game Library tab.
Right click your game in library, and then go to properties. Click on 'Compatibility'
Put a checkmark in 'Force the use of a specific Steam Play compatibility tool.'
Now, as you can see, there are many versions available from the drop down bellow. Which version works, depends on the game. Start with the latest version first. If you still can't launch the game, then go to the next version down. Eventually, one of the Proton versions will make the game run.
BTW: If your ever in the market to buy a new computer, I would be more then happy to help you select one within your budget. Just let me know, and I'll help you out.