The continuing popularity in Iran and the Persophone world of Classical poets such as Hafez and Rumi, and of poetry in general, has no equal that I know of in the West. I suspect that this tends to slow down language change to some extent. People continue to know the old texts because they are constantly exposed to them in vital, modern contexts. Perhaps it's something like the position that Shakespeare and the King James Bible held in Protestant English-speaking lands, though they were widely understood for a shorter length of time (maybe 350 years?) than the Persian classics (700-1000 years and counting, as far as I can tell). I'm afraid that Shakespeare and the Bible are now slipping beneath the waves of comprehension for non-specialists.
Yes, but on the other hand, original lyrics by Hafez, Rumi, and others are commonly used, unchanged, as lyrics for contemporary Persian songs, even for full-on rock songs, dance music, love songs, etc. I haven't heard any songs in the Hot 100 using lyrics by Chaucer lately.
In part, this is probably because old Persian poetry/song traditions were more modern, compared to english poetry from the same period. They had a strong, very old literary tradition. The English didn't, until a bit later. Chaucer was part of building it.
The musical/poetic sensibilities of old/middle English are way too different to ours.
The Bible (especially in Hebrew) can also easily be plagiarized, and you'll find lots of lines/poems that work "as is" in a pop song.
As a cultural reference, poetry is such a big deal that the largest and most popular tv station in Kabul (Tolo TV), operating in the Dari dialect of Farsi, regularly hosts what could basically be described as poetry slams during prime time tv hours.
They have a fairly active youtube channel if anyone wants to take a look.
You should be able to use a site like ganjoor.net, which contains most classical Persian texts, in Persian only, but in modern type (as opposed to simply being a picture of an ancient or modern manuscript page). We've made heavy use of these sites over many years and been very happy. Errors seem to be extremely rare. Keep in mind though, that there will almost certainly be differences, often major, between whatever manuscript you are looking at and whatever modern edition is the one that's been typed into ganjoor.net (or another similar site).
For another approach, there are academic texts to help you learn to read old handwritten scripts. As for deciphering the full meaning (bearing in mind that there may be many layers of meaning and lots of puns and other wordplay), that just takes a whole lot of work. We (my partner in reading for 13 years and I) just start at the beginning of a work (say, Nezami's Khosrow and Shirin) and take it line by line, using dictionaries, grammars (though we pretty much have all the grammar down now), and translations. We use the footnotes, too, in the back of the contemporary editions in Persian. We get the Persian books from Persian on-line bookstores, most of which seem to be in LA. Congratulations for getting as far as you have on your own. Amazing!
Most current-day readers might prefer Davis, but the best in my eyes is that by the two brothers Warner: "Between 1905 and 1925, the brothers Arthur and Edmond Warner published a translation of the complete work in nine volumes, now out of print." https://en.wikipedia.org/wiki/Shahnameh I read classical Persian, I've read most of the Shahnameh in Persian and taught parts of it to English-speakers, and have been amazed by the art of the Warner's work and the depth of their understanding.