(I'm not a native speaker, but from what I've learnt/heard:)
तू is either very intimate, addressing a young child, or rude; तुम is familar - friends, younger family; आप is respectful - older family, more senior people at work or in life generally, or being extra nice for some reason to someone you'd usually use one of the forms with.
Additionally, तुम is the plural form of तू, so using it to address a plurality (e.g. तुम लोग) is like calling any one of them by तू individually.
Adding to this as a native speaker:
तू - To close family members of your age or younger than you and very close friends of the same age as you. It is rude to use this for anyone else. This is the most intimate form but may be seen as juvenile especially in some regions.
तुम - To almost anyone who is not older than you when you're not being formal. More intimate/less formal than आप.
आप - To someone older or when you're being formal. Note that it is not seen as nicer to use आप with someone you usually use तुम with (it may be seen as distancing yourself). The order should always be आप (when you've just met the person) to तुम (when you become close to them).
तुम is not the plural form of तू. It's just that तू can be used only when you're addressing a single person while तुम and आप can be used to address one person or more.
Well, if you are talking informaly then the use of tu/tum would be fine.(tu is more little, very little bit more casual.) They are used while talking to someone younger or same age, i.e- friends. Aap is used when you have to pay respect or in formal speech. It used with elders normally. I.e- your parents, teaches, elder brother/sister, someone at a higher post than yours, etc. Hope it helps!
‘her’ in this example shouldn’t be capitalized. In fact, it’s too easy to identify the first word in these types of prompts when they have a capital letter. The word block could be all in lowercase but present the correct solution (in green bar) could still show the correct capitals for reference.
No. Pluralisation can be used to convey respect in the third person.
उसका पिता - regular form (may be considered rude)
उसके पिता - respectful to the father
उनके पिता - respectful to both the father and the son/daughter
उसकी माँ - either regular form or respectful to mother depending on whether you conjugate it as singular or plural
उनकी माँ - respectful to the son/daughter (conjugate it as a plural to be respectful to the mother too)
हैं is the form of होना that conjugates with आप.
However, in conversational Hindi, you often see 'आप .. हो' as a form that is intermediate in formality between 'तुम .. हो' and 'आप .. हैं'. Another forum member had mentioned that this may be due to the influence of Punjabi.