Since Intel is one of the few companies that actively support open source drivers for their GPUs, they are probably the most robust GPU you could use under Linux right now because of that alone. Personally I would buy an Intel graphics card just for that, but sadly nobody makes video cards with Intel GPUs.
Dual monitor support depends on the connectors on the machine itself. I have seen some machines with recent Intel GPUs that can support two digital (DVI/HDMI/DisplayPort) screens plus one analogue (VGA) for a total of three screens. It just depends on what connectors the manufacturer decides to use.
You'll need to check the specs, but most video cards will output up to 2560x1600 over their digital connections, and 2048x1536 over VGA.
The performance of the GPU doesn't decrease as it's logically separate to the CPU, and power usage is based on the type of GPU rather than whether it is integrated or not. Typically dedicated GPUs are faster, so they use more power. Possibly an integrated one might use a tiny bit less power than an identical dedicated GPU as it may not need as much supporting circuitry, and probably won't need a separate fan as it can share the CPU fan, but the difference would be pretty small.
If you're not gaming and you're running Linux, the advantages of an Intel integrated GPU are pretty clear: Solid open-source driver support.