Ok thank you for providing me with my evening's entertainment.

1)They had a pretty major impact on art. However this was not so much in terms of them introducing Mongol motifs into art, instead they brought in motifs from other areas they conquered and came into contact with. So in Persian paintings we see landscapes painted in Chinese styles. Another good example is the Universal History/Compendium of chronicles which was compiled by Rashid al-Din, a Persian serving as a government minister for the Mongols. This manuscript was rather beautifully illustrated, and we can see a wide variety of styles influencing the art such as Buddhist, Siennese Italian, Chinese and Byznantine iconography. If you look at this and this, both from 1314 copies of the work, you can see that the illustrations have some rather Chinese influences. These manuscripts would would have been illuminated in government workshops in Tabriz which had artists from across the Empire working in them.

It's also interesting to note this influence was not one way. There's a good article called Siena on the Silk Roads by Roxxan Prazniak which examines the influence of Ilkhanate art on Siennese art.

2)This question is more difficult as they don't appear to have left many distinctly Mongol traces. The Mongols were fundamentally a very adoptive empire, although in China you do see some moves to try and preserve the Mongol identity such as banning intermarriage. However in the Ilkhanate there were no comparable measures. After about 50 years after the Ilkhanate is established the seventh Ilkhan converts to Islam. This sees the end of the rather interesting introduction of substantial Buddhist communities into Iran. But I'm getting sidetracked. So basically we don't see huge influences overall.

Now for a while there was a universally accepted theory that the Mongols introduced their own system of laws known as The Great Yasa. However this theory has been pretty much debunked, all the supposed references to it turned out not to really to mention it in the end. It was likely a compilation of Mongol customary law or Genghis Khan's sayings/advice.

The one area off the top of my head where I can think of the Mongols having a clear influence over the status of women. The Mongols on the whole were pretty flexible about giving women authority, out on the step you couldn't isolate women in the house they were needed to do things. As a result they had a fair amount of freedom and even had a degree of power over men. Thus as a result Mongols were pretty okay with giving women political power, I could go on about Torgene but this about Iran. Some good examples in Iran are Terkhan Khatun a woman who ruled Kirman as a Mongol vassal, although she also benefited from her dynasty's Central Asian background which meant they were also looser on female confinement. That said this was likely confined to the elite sphere as the Mongols did not try and interfere with the lives of its everyday citizens on points like these. Furthermore after the Mongols left this liberalism did not really last. Interestingly I've seen some scholars suggest you can chart the decline of Mongol influences by looking at women in politics.

On Iranian culture making it back to Mongolia I'm afraid I'm far weaker. An obvious trait, Islam, did make it back to Mongolia. William of Rubruck a Papal envoy who ended up in Karakorum, the Mongol capital, noted two mosques and later engaged in a debate with some Islamic scholars, and some Buddhist ones, in front of the court. However Islam didn't meet with much success in finding converts in Mongolia, although Mongols in other areas did convert.