1: Can a cheaper cnc router like the ones made by Cheapcnc cut 1/4 Lexan or 1/8 Carbon Fiber? I do alot of custom work by hand but would like to start using cnc for it.
It all depends on how fast you want to cut your parts. A very weak machine can cut what you ask, you just would need to take very shallow cuts and slow speeds.
2: Can there machines do drilling as well?
Sure. The main problem would be the RPM of the router and finding a drill chuck that will handle the high speed. I tryed drilling with my vara-speed router on 10,000 RPM. It worked but shook a bit.
3: Is it worth it to just buy the parts myself and make one or would that be too hard for a beginner? If so, how much will it cost?
It is not hard to build one. I see you can run a CAD station. You need to make the choice between router or mill. The part you have shown could be made on my converted Mill/Drill (see my website). Also I just wanted to say that using a router or mill, you will not be able to cut square corners on pockets like you have drawn. I would say to not be scared to build one or to convert a mill/drill. You will have all the help you need from us here.
Thanks
Jeff Davis (HomeCNC)
http://www.homecnc.info
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)