Is it true that running an AMD or ATI GPU on an Intel CPU will actually make you lose performance?
I believe this is invalid. They should perform the same on different CPU architectures (provided they're up to date.)
Since there is no other place to ask, is a Radeon HD 6850 graphics card fully compatible with my Dell inspiron 518 desktop?
I'm assuming your processor is a Core 2 Duo.
Provided there's enough room,
and an adequate power supply, yes. I just looked at the documentation a bit further, and there's only a 300W PSU in there. You'll need to upgrade that too.
Keep in mind though, you may experience slight performance drop compared to some other people, as your motherboard is only PCIe 1.0A.
PDF Documentation here.