As of quite recently (24-02-2015) it is possible in Google Documents to have a different header/footer on the first page. Therefore it is only possible to fulfill the second instance you described.
Text posted on the Google+ Google Docs page:
Headers and Footers - the way you want them
One of your biggest requests (especially those of you who use Docs to write
academic papers) is to be able to customize how headers and footers appear
on the first pages of your documents.
With today’s updates, you can now use different headers and footers on the
first pages of your documents, which comes in handy for:
> Following academic formatting guidelines (e.g. MLA, CMS, APA)
> Starting page numbering on the second page
> Making a title page without headers or footers
After inserting a header or footer in your document you’ll see a new checkbox
--just tick it and you’ll be able to set a different header and footer on the
There’s also a new Insert > Page Number menu that lets you customize how and
where you start your page numbers--like on the bottom of the second page,
It might take a while for the feature to be available as it's being rolled out.