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That's right; outside of the TCG, those which you call subtypes are Types in their own right and always were, which takes us back to your question: yes, a pokémon can be of any type/"subtype", or a combination of two types in any order, the first one being the "main type" and the other the "secondary type". Flying is an exception as noted, always going in second place combined with a different type. No pokémon is pure Flying-type naturally, only through the use of special means which are mainly temporary.

Another special case is the Normal-type, which is hardly ever seen combined with other types. When it does, it is usually paired with the aforementioned Flying-type. The only exceptions so far are Girafarig (Normal/Psychic) and Bibarel (Normal/Water). Based on these facts, I believe that just like Flying is a Secondary Type by nature, Normal is conversely a Main Type by nature.

No, you forgot another one: Azurill. It goes from Normal to Water upon evolution into Marril, without the need of an evolutionary stone like Eevee. Now the key question is: what makes Bidoof different from Azurill? Why doesn't it just evolve into a pure Water-type as well? I think the answer is that Bibarel is less aquatic than Marril, and it certainly looks like it to me. Nothing in its design would cry 'water' if it wasn't for the fact that it's a beaver. In short, I fully support the Normal/Water combo for this particular exception.

Going back to your question, PCZPhyre, as Poryhedron said there is no official pokémon with those combinations, so we technically don't know if those types could be combined or not. But since Normal combos other than Normal/Flying are rare, our policy towards them is restrictive: there has to be a very, very good reason for the pairing. So far, the only new combination that PF has introduced is Normal/Ghost, for pokémon that keep a tangible or semi-tangible animalistic form.

Yes, look at the top of page 2.