Not unless you are from the 19th century. In modern language thou shalt not use polite forms with members of your family and with kids.
You might recall that some languages have the Vocative case: the form of the word that you use while addressing a person (by their name, title etc.) Russian lost it long ago but, amazingly enough, in contemporary Russian forms of Neo-Vocative arised for informal names and for very few nouns (мама and папа, essentially). The last vowel sound gets removed:
- Александр(а) → Саша → Саш
- Иван → Ваня → Вань
- Пётр → Петя → Петь
- Алексей → Лёша → Лёш
- Мария → Маша → Маш
- Елена → Лена → Лен
- Екатерина → Катя → Кать
- Сергей → Серёжа → Серёж
- мама→ мам
- папа → пап
These are optional, and the system is nowhere near well-built (I mean, why only a few dozen names have it?) but they do act as a Vocative, while having nothing to do with Russian historical Vocative (still crystallized in Боже! "God!")
It is beyond the 6.5 cases we teach here. Neo-Vocative is optional, though you will hear it used sometimes. So it's useful to know your ears did not fail you when you thought you heard «Эй, Саш!».
Not a great analogy. "Thou" is the ты equivalent in Early Modern English.
It would sound kind of funny in russian (especially with a mother), cause we don't use words from foreign languages that often. But actually you could use 'сорри' instead 'извини' in an informal situation, for example with your friends.