May I ask which laptop you have at work ? I don't mind 14" screen. I'd buy a second screen eitherway ...

It's one of the Gigabyte P34 range. It was about USD$1300 new or $800 second hand.

I just went through Baylor University using 17.3'' MSI gaming laptop (was easily able to handle the graphics/engine courses, etc.) and it cost around $1200. Has an integrated graphics card as well as an Nvidia GTX970M (which was WAY more than I actually ended up needing). Don't know if that helps you much.

Anywho, was just gonna suggest that, as for the notes, you may wanna try grabbing a rocketbook. They'll be releasing over the summer and it lets you take notes on paper (ensuring you pay attention in class like the studies frob mentioned), but also lets you upload the images to cloud services and then clear all the pages by microwaving the book for 4 minutes. Pretty nifty concept.
https://www.indiegogo.com/projects/rocketbook-wave-cloud-ready-microwavable-notebook#/

Well, small screen size can be better for portability but also have to keep in mind about cooling infrastructure which can be problematic in smaller ones. On my own I'd recommend a 15.6" laptop, my must have is IPS screen but might not be crucial in case of extra monitor.

I am keeping an eye on Dell 7559 laptops nowadays but not sure if available there or price is fine.