As far as the HSMP, the programme is actually changing right now as we speak. It is suspended for a few more weeks and then the rules totally change. It is really important you understand the changes. Take a look at the Immigration board here for some advice as well as Victoria, who is regular on our boards or another immigration consultant would be best to talk about your HSMP qualification status. With the rules changing rather dramatically.
As far as the HSMP, you require no sponsorship at all, where an H1-B does. This would be similar to someone having a Greencard in the states. There are only a few Civil Service jobs that you cannot apply for and I find that if you have enough to get an HSMP that most employers would find you attractive. Being as there is nothing they need to do, they will put you in equal running with other candidates.
Generally it is all UK checks, and varies greatly by industry and career. Credit checks are not quite yet standard fair like they are in the US. There was a recent posting on here about someone who applied for a UK job and the company wanted to run US and UK background checks. Lots of jobs that deal with children or the disabled will require a criminal record check in every country you have lived in.
The lack of UK residency tends to be more of an issue with credit and and getting a mortgage or even renting, but doesn't tend to get in the way of getting a job. I am personally finding, again if you are skilled, that most employers just need a little education in how your visa works, so you just need to make sure you understand it.
Any resident of the UK who is eligible to stay and plans to stay > 6 months is entitled to NHS (healthcare) the day the enter the country. This would include HSMP. As far as government pensions, unemployment, etc. are a rather complex issue because of agreements in place with the US. Taxes become a rather ugly thing as well, because the US is interested in your worldwide income and will want to tax anything you make over $80K per year. Something you should consider before taking the leap.
There are no issues with the spouse working. They are free to work as they choose as long as they have the proper dependent visa.
Welcome and take a look around, these boards are filled with invaluable information on the whole process.