Does writing influence grammar?

This is really not the right question to ask. Writing is not just about the way of writing things down but also its own way of communication. People who read and write, don't just transfer speech into letters. They express themselves differently in the two modes. This is partly because of the different cognitive demands of reading/listening and partly because the two are used in different contexts. But in effect, all literate people are to an extent bilingual - they are competent in two codes and they switch between those codes with ease.

But as in all instances of language contact, there is cross-pollination. So it's quite likely that complex multi-clausal sentences with embedding and hypotaxis developed in complexity alongside writing - although I don't think there is any definitive proof of that. In some cases, written language perpetuates older forms - e.g. in Czech or Arabic that would probably otherwise disappear. In English, there are many pronunciations influenced by spelling and vice versa. None of these examples constitute 'imperfections', just features. Although, it would not be difficult to argue that the English spelling system is completely flawed, the impact of these flaws on spoken English has been minimal.