.gov is the domain for the US government.
.gov.uk is the domain for the UK government.
Commenting that something "is preferable for domains" seems inappropriate, because the domains have already been determined and it's merely a case of using what exists. Preference does not come into it.
My choice of whether to capitalise the initial letter of the abbreviation would exactly follow whether I would capitalise the initial letter of the full word in the respective context.
A domain name extension would, of course, never be capitalised.
[The comments below apply equally to capitalised and non-capitalised versions of the respective abbreviations.]
I have always understood that conventional practice on whether to include a period at the end of an abbreviation is to include one if the last letter of the full word is not present in the abbreviation, and not to include one if it is present.
Hence I would omit it from Gov't and Govt,
but include a period in Gov..
I doubt that there is any standard on whether to include an apostrophe in the abbreviation Gov't, nor any consistency within any particular government, let alone within governments from different English-speaking countries.
My personal preference is normally to include one.
t or no t
Although I don't have anything off-hand to back this up, I would suggest that inclusion of the t is possibly more common (in the UK), and that the abbreviation gov. without the t has become more prevalent from its usage as domain name extensions.