I hate plastic kitchen counters corian or quartz. Yes you can get integrated sinks with corian . IMO a good quality deep single bowl stainless sink will always be my go to . Any sink material that can stain or show hard water stains would not work in my life .There is a reason stainless is used in commercial kitchens it takes ahuge amount of abuse and still cleans up to look awesome

Thank you Patricia! I actually love the newer quartz materials ( to each their own) but do agree with you on a good quality stainless steel sink.

"Can you do an undermount sink with Corian or do intergral sink?"

Yes.

"Anybody have any experience with either?"

Yes, and as a Certified Corian Fabricator/Installer, it's extensive and in addition I've retrofitted/replaced many different sinks into Corian.

"What brand/sink would you reccomend?"

This is kind of like asking "What car would you recommend?" Every sink and sink material has tradeoffs; only you can sort through those. If you like stainless steel and Corian, I'd highly recommend the Karran 540:

No clips, no straps, no caulk, no reveal, no maintainence, just a seamless transition from top to sink. If there is a more easily cleaned or sanitary sink/top, I haven't seen it.

You can undermount an apron sink into Corian. Slight positive reveal please, second only to the above:

If you're into composites, I highly recommend the Elkay Quartz Luxe line:

"Do Corian countertops come flush to the back wall ?"

Yes, but I'm not sure I'm understanding your question. Corian is easily scribed exactly to wavy walls.

Thank you Joseph! I guess I stated that incorrectly. What I mean is I would like counter top to meet the back of the wall then have tile from where the counter meets the wall .. if that makes sense. The 540 looks nice but it being 18 gauge is somewhat concerning especially for kitchen sink that takes lots of abuse.

An 18 gauge sink is plenty adequate for kitchen use.