Yes. As I said in most cases postcode and house number are enough. There may be odd exceptions but it's rare for a postcode to cover more than one street.
It is indeed not common, but it certainly happens. Similarly, a number and street name and postal district is - very occasionally - insufficient without a district name too.
Under international postal conventions - which all national postal administrations used to sign up to, and presumably still do - a "valid address" is a set of information, moving from the general to the specific, that would allow premises or a letter box to be uniquely identified. Most countries do this "back to front" - ie the specific first (ie house number, then street name, then district, then town, etc etc) - but in a few places it's different, where the convention is to start with the city and "work down". The key thing is that the information is logically structured and uniquely defines the premises concerned. Hence in this sense - and logically anyway - a postcode is not essential providing the rest of the information is clear.
Post in the UK is frequently delivered successfully without a postcode - or indeed with an incorrect postcode. (The latter since post is delivered by human beings who can read, not [yet] by machines.) Logically, the postcode is something derived from the address to assist the postal service's automated systems. However, in an age of computers and surveillance, it is being increasingly put to other uses for which it wasn't initially designed. This is exacerbated by online systems which try to deduce your address from your postcode, rather than the other way round; I have personally experienced situations where doing this produces an address which might
work, but which is misleading in terms of the actual physical address (ie the traditional "hierarchical" description of how to find a place), and indeed produces an address in a form (such as the name of the street the premises is on) which has never been used by the occupiers.
Without wishing to muddying the waters a postcode can be assigned to a business or organisation. Mail addressed with that postcode is then aggregated and delivered to an address which may have a different postcode. This is to facilitate a form of sorting for the business.
Indeed - I know well a building where one (historically large user of the postal system) was allocated one postcode, and the rest of the building (and the rest of the block) was allocated another one. Hence this building (though only a small shop+offices terraced building) has two postcodes, one unique and one shared with the block. Providing the building number is correct the post gets through irrespective of which postcode is used, and whether or not the postcode is the "right" one for the orgnanisation concerned.