Not necessarily. It is not unheard of to refer to your close elder relatives with the tum-level of formality as a show of endearment. This is especially true for your mother or other close female relatives.
Of course, if the sentence with हैं is not also accepted as a correct answer, you should report it.
Tum does conjugate with ho.
But usually, for people you call tum or tu, when you're talking about them in the third person, you conjugate the words referring to them as singular. If you use aap in the second person, you conjugate the words referring to them in the third person as plural.
(This is not always true. For example, who you are talking to may sometimes also be a factor. But it is a good thumb rule. )
if the sentence with हैं is not also accepted as a correct answer, you should report it.
Except on a "Type what you hear" question. For those questions you need to transcribe what is actually said, not a response you composed yourself, even though your phrasing would have been more respectful. :-)
"लिखना" (to write) is the base form of the verb लिखता "write/writes". I don't know whether the "writing" that you're saying is a continuous verb or a noun but the continuous form of the word "लिखना" will always be "किख रहा/रही/रहे" Eg. मेरे पिता बहुत तेज़ लिख रहे हैं (My father is writing very fast)
Also, the noun "writing" are either "काख", "लिखन", or "लिखाई" so I really don't know what you mean.