When I use Alt + Shift to switch from Minecraft to Chrome, the screen goes black for a few seconds, which is probably due to the different resolutions. Is there any way to change the resolution of the chrome window?
In theory, yes. That would be quite complicated and not really good.
If you absolutely want to play with a lower resolution and still have Chrome open in the background, you should rather change the general resolution. You can set this either in the NVIDIA Control Panel or the Intel HD Graphics control panel.