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Will the power transfer thru the splitter,
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Only if you buy one with DC pass through. Any of the ones in retail stores rated for satellite service will have this feature. I'd suggest the Ideal branded splitter available in Home depot stores, but you could probably use anything from Walmart, Radio Shack, Lowes, Menards, etc and it will do the job.
Your selected preamp has more than enough gain for your requirements.