stupid question maybe but are those the only two options or are there more?
That's all the imperative forms there are. The "ты" form is informal and singular, the "вы" form is formal or plural.
OK. Was really confused by that. Thanks for the heads up. Much appreciated.
Why is 'Извините, папа' wrong. Is the polite form never used to adress parents?
Polite forms used to be common with relatives and even parents in the 19th century (at least, in literature, so one might assume that it was common in some circles). In modern usage, вы-forms are not used within one's family.
No, not normally.
Edit: It's maybe oversimplification to say that вы is the polite/respectful form. It's used to create distance between people, from respect or politeness or some other cause. Using ты shows closeness, so you use ты with parents.
Is there any difference between прости and извини? Is there also the вы-form of the first простите?
I would also like to know the difference between простить and извинить, in general. Is there a difference in politeness or connotation between them? My impression is that простить can also be used in the sense of forgiving, rather than just "sorry" or "excuse me", but I haven't found anything confirming this.
Duo has suggested Пап as an alternative to my папа. Someone up above suggests it might be analogous to US "Pop". Is this true, and if so, is it common?
Пап is an example of what's called the neo-vocative case, which can be used when addressing the person. The meaning is no different from папа. How common it is, I couldn't really answer. I believe it's pretty common among younger people, certainly in my limited experience.
I live in the US and I've never heard anyone call their dad "pop" in real life.