Yes you can, if this is what you mean by angled connectors :- http://tinyurl.com/dyouoo
BUT, it you intend to have all you interconnects and mains cables running through that small hole you'll be in trouble. The mains cables, unless very,very well shielded (most aren't!) can induce hum into the signal cables and the signal cables can induce noise into each other, especially if they're running a low level signal such a turntable lead.
Can't you make another hole in this cabinet and run the mains cables through one and the signal cables through another?

Cutting holes in the back of the cabinet would be ideal, if that's an option.

Yes. Cut big holes. Or remove most of, or the entire, back panel.