With the healthcare bill as wildly unpopular as it is, does the House of Representatives have any business passing it? Or anything else that gets everybody up in arms?

AFAIK, the House is supposed to represent the citizens of the various states, and reflect their collective will. The Senate is supposed to represent the states themselves.
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Disclaimer: This is NOT about whether or not we should have Obamacare; there are already discussions enough about that. This question addresses procedure, policy, and practice.