Would it be possible to have Steam run linux ready games natively, and non-linux based games through wine without having to run two instances of Steam?
Now is posible to connect two clients (Linux-native and Wined-steam) in same machine in same os.
The dual login is now possible because of Steam Home Streaming feature, the restriction login changes and life goes easier for us.
*NOTE: IDK if Mac users can use this feature, but Linux user can use Home Streaming, i use it daily the webpage is outdated (10/22/2014).
Even if it could be technically possible, you can't be logged on the same Steam account at the same time in two different clients, and I imagine that you wouldn't have two accounts with games split around.
Running games on wine would also be game-dependant more than steam-dependant since each game has it's own requirements and prerequisites, therefore some might work, some might not. A better solution would be running dual boot, or a virtual machine, with both linux and windows, but I wouldn't rely on a virtual machine's performance.