Can I develop an XNA 4.0 game for the older Xbox 360 Elite? Or would I need a newer Slim model?
The development platform for all Xbox 360 models is identical - that's the nice aspect of developing on consoles. The models differ in mostly incidental things like disc drive speed, number of cooling fans, and hard drive size.
The speed of the hard drive and disc drive are the only things that even remotely affect game quality / technical requirement compliance, and unless you're streaming lots of data, they're unlikely to matter for you.
Any Xbox 360 will do, regardless if old or new since they are essentially all the same in terms of CPU/RAM/GPU.