Is the United States Government a proper noun? United States is, of course capitalized, but is Government also proper and needing to be capitalized in this case?
Yes, "Government" should be capitalized in this instance.
If you were saying "The United States Government should do...," then you would capitalize "Government." But if you were to say "The government of the United States should do...," then "government" wouldn't be capitalized because it's not actually part of the official title. So if it's part of the person, place, or thing's title (Aunt May, etc.), then you capitalize it.
No. Government is not a proper noun here.
In this expression "government" is a common noun. It does not describe a specific institution.
An official name would be proper noun and as such should be capitalized -for example: The Secretary of State (of the Untied States).
Other example would be word like parliament. Although it describes a speciffic institution it is not a proper noun. So expression like English parliament is correct. Only the official names of (both) houses are proper names. So only correct forms are: House of Commons or House of Lords