are these words commonly used？：很好 、极好 、非常好 、相当好 、特别好 、十分好、极其好
Let's forget about '好', you can replace it with any other adjective. Your question is about the difference between superlatives
很 = is; very
极 = very; extremely
非常 = very; extraordinarily
相当 = quite; very
特別 = very; especially
十分 = very (fully)
极其, 极之 (more colloquial) are the same as 極 (more literary)
Examples:
她的成绩很好 = her grade is good/ her grade is very good
她的成绩极好 = her grade is very good/ her grade is extremely good
她的成绩相当好 = her grade is quite good/ her grade is very good
她的成绩特别好 = her grade is very good/ her grade is especially good
她的成绩十分好 = her grade is very good
More examples:
他跑得很快 = he runs fast/ he runs very fast
他跑得极快 = he runs very fast/ he runs extremely fast
他跑得相当快 = he runs quite fast/ he runs very fast
他跑得特别快 = he runs very fast/ he runs especially fast
他跑得十分快 = he runs very fast