I think the exact method of writing addresses depends very much on local custom. In the US, UK, Canada, and Australia, the normal form is something like this:
522 3rd Avenue #4
- the street is "3rd Avenue"
- the house number is 522; that is, this is the 522nd building on 3rd Avenue
- the number of the apartment (or flat, or unit) within the building is 4.
We don't usually say or write out "number" or "No." for the house number except in a few famous cases like Number 10 Downing Street (where the executive branch of the government of the UK is headquartered).
The issue in your case arises because in English-speaking countries we don't usually begin street names with a number! If the street name itself is a number, like 3rd Street or 5th Avenue, we usually write it as an ordinal, like 1st, 2nd, 3rd, etc.
To answer one of your original questions, if the street is really named "15 Alpha Street", then no, I would not assume that there are multiple Alpha Streets. The US has a few "4th of July" roads and Montevideo, Uruguay has streets like "18 de Julio"; I don't think anybody thinks there is also a "5th of July" road and a "19 de Julio" avenue.
But if the street name itself begins with a number, like "15 Alpha Street", then something like
20 15 Alpha Street
becomes very confusing, because we're only used to seeing one number before a street name, and that number is the house number. If there is only one "15 Alpha Street", then I would write out the address like
No. 21, 15 Alpha St