Recentemente (24-02-2015), no Google Documents, é possível ter um cabeçalho / rodapé diferente na primeira página. Portanto, só é possível atender à segunda instância que você descreveu.
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Texto postado na página do Google Docs no Google+:
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
first page.
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,
for example.
Captura de tela
Nota
Pode levar um tempo até que o recurso esteja disponível enquanto está sendo implementado.
Referência