My boyfriend who is Italian has told me that there is no difference between "fa tardi" and "è in ritardo", except that you can also use "fa tardi" to mean that someone stays up/out until late. So this sentence could also mean my mother stays up/out until late.
"è tardi" is impersonal and cannot be used when the subject of the sentence is a person - it is used in the form "è tardi per + verb" (it is late for ___). If you want to say someone is late it is always "fa tardi". If the subject of a sentence is an object (e.g. the bus is late) you can use "è tardi" but it is more common to use "è in ritardo".
Hope that makes sense.
Because "madre" = mother and "mamma" = mom/mum. I also got sentences marked as wrong because of this. I think(this is how I understood) it is because we need to differentiate between the formal and informal terms? I also saw an explanation on another sentence but can't remember which. I hope this helps.
Unfortunately, I'm not answering your question (which I would have asked as well, re. e and fa,) but could it be that we have to read it as "my mother 'does' late", as in "my mother is always late" - as that is what she does? It did mention that it was used when it's a planned lateness.
The definite article is not used with family members unless the noun is plural or modified by an adjective or the possessive is 'loro'. So 'mia sorella' but 'la mia sorella preferita,' 'le mie sorelle,' 'la loro sorella.'