Regarding the constitutional problem with Hillary Clinton moving from the Senate to the State Department...
Article I, Section 6 -- No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil office ... the Emoluments whereof shall have been encreased during such time.
It's standard operating procedure for Congress to pass a law, however, that will allow a senator who voted for a cabinet pay hike to subsequently serve in that cabinet post. The workaround is known as the Saxbe Fix, named after Nixon attorney general William Saxbe. Basically, the new law returns the salary back to its previous level. Naturally, it doesn't reverse the fact that Senator Clinton did, in fact, vote for the pay hike.
No doubt, it's one of those sausage-making issues that looks really, really fishy, but ultimately never amounts to anything.