Designing USB device
This is not a complete answer, and your question is too broad for that. I can give you my experience.


  Are there any reasonably priced eval/dev boards for USB chips?


I can't answer this.


  Can you recommend any specific chips?


I have used the PIC16F1454 in two projects. It has everything built-in for use as a USB peripheral. You literally only need 1-2 capacitors and a USB connector. It has a built-in 3.3 voltage regulator for the USB drivers, slew rate limiting, pull-ups, everything. Runs on the USB 5 volt.

(A lot of people will likely scream at me for recommending a PIC - deal with it. I hate it as much as anyone, it has a horrible architecture internally, but who cares today, you will code it in C anyway. Considering that Microchip is buying up Atmel now, it's also hard to tell what's going to happen to the AVR controllers.)


  I'd like to learn more about usb, I've researched a bit and I'm a bit confused


You're not the only one who's confused. USB is hard. If you want an overview of USB I can recommend USB Made Simple.


  Would writing my own usb drivers for my device be too crazy?


When you write "usb driver", I assume you mean on-chip firmware. It is not crazy, it's just stupid. A normal embedded developer will use one of the USB stacks supplied by the manufacturer, or by someone else who was crazy enough to make one, essentially picking a program that's close enough to whatever you want to achieve. For the PIC16F1454 I recommended, there are hundreds of example programs with corresponding client software, or there is the open source M-Stack.


  Is there anyone else who done something similar that could share their experience?


Likely, but not on stack exchange. Or actually, that is what I'm doing right now, in a limited sense.


  Should I ditch my plan entirely and just get some sort of an off-the-shelf solution?


No, because you write "I want to design that peripheral device from scratch i.e. create my own PCB.". This is a nice goal and your question is a good start.