First question: If you're working for a major, unique employer, then use "the". So if you're working for the UK Home Office government department, it'd be
I am working for the Home Office
The same thing would be true if you were working for an embassy:
I am working for the American Embassy
Even though there are lots of American embassies around the world, it's still
more or less unique in the country you're in, and
I am working for American Embassy
would definitely be wrong.
On the other hand, if you were working for a company called "Home Office", it'd be
I am working for Home Office
since there are many companies.
For your second question, as a native British English speaker, either works, though I'd probably go for
Employer: Home Office
since it's not part of a sentence.
For the company with the same name, it'd be
Employer: Home Office Ltd (or Home Office Plc, or whatever)