This is my exact area of study! Which actually makes this a little bit of an overwhelming question for me to answer because I don't want to just paste my whole dissertation bibliography here. There also isn't a lot of stuff I'd wholeheartedly recommend as having absolutely zero issues in it, and especially not anything that's in any way comprehensive - the high-quality published work tends to be pretty specialized and doesn't give you a good big picture view. For big picture looks at topics and periods, your best candidate to try is the Cambridge History of Iran. Also, there's very, VERY little on average people. But here's where I would say to start:

Anything by Shahzad Bashir is going to give you a good sense of the quirkiness of the religious environment, and will be quite readable. My personal favorite is Fazlallah Asterabadi and the Hurufis.

For visual art, which I highly recommend spending a bunch of time on because it's such a magnificent feature of the period, try anything by Sheila Blair, or Beyond the Legacy of Genghis Khan ed. Linda Komoroff, or (STRONG REC) Timur and the Princely Vision ed. Lentz and Lowry. This is a gorgeous book based on the exhibition that kickstarted modern Timurid studies in the English-speaking world, and the articles in it are really solid. It also has a lot of material culture in it, not just painting, which is great. Oh and you can never go wrong deep diving on the Met museum website!

My most specific recommendation other than Lentz and Lowry is a paper by Maria Subtelny: "Scenes from the Literary Life of Timurid Herat." This has a really vivid primary source account of a 15th century party. I also would say that the strongest answer to what created "a shared sense of culture" is really "Persian poetry," and this party features poetry. For more on that, Annemarie Schimmel's A Two-Colored Brocade gets deep into how poetry worked culturally.

In general, travelogues are my favorite primary sources on mass culture because travelers often take note of things that people who live in the place take for granted. William of Rubruck, Ruy Gonzalez de Clavijo, and especially Giosofat Barbaro are my favorites for this period.

I'm afraid I can't offer much on economics or demographics. I'm not even really sure what resources you'd use to do high-quality work of that kind on this period, since there are no surviving archives... there's some stuff on numismatics/coins, so let me know if you're interested in that.