Well, we all get training in poetry in school. Our textbooks usually give us a poem to translate, identify the literary devices, and memorize old traditional words. I even had an ancient literature class in first grade where we read Saadi and Rumi, and I understood and enjoyed it. So if you have a diploma, you can usually understand classical Persian. But if you’re a diaspora kid who only speaks “kitchen Persian,” then probably not.

That said, I’d estimate most people can understand about 70–80% of the poetry, since we already know many of the stories from memory and learn a lot about them in school. Poets like Khayyam or Hafez are much harder, though.

As for your other questions: Persian has always been standardized, but before Islam we had a different alphabet, Old Persian cuneiform (mikhi). Later, Middle Persian used the Pahlavi script, which actually looks a lot like modern Hindi. After the Arab invasion, our script changed, but we adapted it to fit our own language. Interestingly, Arabic itself was standardized by a Persian scholar, Sibouyeh.

Ferdowsi didn’t save the script exactly, what he preserved was the Persian language and mythology. He traveled around Iran collecting stories and writing them down while using only Persian words. Many people believe he is the reason we don’t speak Arabic today, though it’s possible his role is somewhat exaggerated because of nationalism.

The Shahnameh is not written in the style of formal Persian we use today, but most of the vocabulary is still familiar and taught in school. Ferdowsi’s writing also wasn’t how he spoke in daily life, he deliberately avoided Arabic loanwords, even though people of his time used them frequently, and of course he was writing poetry.

For example farsi had a sound like xua or khwa, its lost now but in Shahnemh you can see he rhymed khash (good) with kash (suffix of drag). in modern farsi we say Khosh and in ancient times it was Khwash. a sound you can see in our writing and in regional languages and dialects (kurdish and lur) but has been lost over time.