Is the repentance of one who insults God or religion valid if he repents of that, but he is still persistent in insulting people? Is this considered repentance from a sin while persisting in another of the same type? And what is the ruling on repeating songs that contain blasphemous words?

Repentance from a sin is valid even while persisting in another. Insulting religion is an act of disbelief (kufr), and repentance from it is valid. Through this repentance, a Muslim becomes a Muslim again, even if they have not repented from insulting people. However, it is obligatory to repent from insulting Muslims, because "insulting a Muslim is a transgression (fusuq) and fighting him is disbelief (kufr)."