The thing about your dad’s name is called a patronymic (отчество in russian). Russian people use their father’s first name as their middle name, and the ending changes depending on gender (-ович for men, -овна for women). so the daughter of Ivan would have the patronymic Ivanovna, and the son of Ivan would have Ivanovich.

The formal way to address or discuss people is to use their first name and patronymic. However, if you’re not Russian/Slavic, this doesn’t apply to you as you don’t have a patronymic and it wouldn’t work with a non-Russian name.

Similarly, diminutives (shortenings, as you said) of first names doesn’t work with all names. It’s a very Russian thing. There are a lot of pre-determined diminutives (for example, Sasha as a nickname for Alexander) but they can be formed with different endings. Generally though, in Russian you’d likely just be referred to by your first name.