Translation:A family with small children lived above me.
Why does an "e" get added after nad? I noticed the same thing with "beze otce".
It is "nade" in front of a group of consonants. Here "mn" is the consonant group.
If you saw *"beze otce" anywhere, it was wrong, because "otce" begins with a vowel. "beze jména" is the right example.
However, it is not that simple, consonants l and r are different (e.g. they can by syllabic) and may not count. And sometimes the given combination of words is just the variant native speakers are "used to". You have to remember those.
Examples (just words that quickly came into my mind):
bez vražedných úmyslů
beze skvrn / bez skvrn
beze srážek / bez srážek
Often you can find both options, "bez ..." and "beze ...", in the korpus (https://kontext.korpus.cz) and one of them is just more common. I didn't test all the above in the korpus and I do believe than often both versions are possible although I personally preferred one of them.
Very good explanation. I have 2 remaining questions. 1) Is this more of a spelling issue? It seems you would just naturally add the "e" in the correct situation to make it easier to pronounce the phrase. 2) Does this apply to ALL words ending in consonants prior to a consonant cluster? Or only prepositions? I know "strč prst skrz krk" doesn't have any added vowels!