1. The mathematical objects are "Zahlen", i.e. your kids learned the Spanish "Zahlen".
2. a) "Nummern" can be used to identify an object within a given order. That's why we have "Hausnummern", for example, or can refer to the third person in a queue as "Nummer 3", or a doctor could say something like "Mr X ist Patient Nummer 3".
b) "Nummern" can also refer to a string of digits used as a mere identifier (without any direct idea of order involved): digits referring to different products of a company, "Artikelnummer", digits referring to a particular lottery ticket, "Losnummer" , or to a flight, "Flugnummer", for example.
What a) and b) have in common is that the numbers are not used as mathematical objects, i.e. you do not count with "Nummern", not do you add them up, for example.
3. Very colloquially, "Nummer" can be used synonymously to "Zahl". But, depending on the circumstances, this might sound extremely relaxed or even funny.