In 13.5, the youkai made from the original collection of Noh masks made by Prince Shoutoku, Hata no Kokoro, lost her mask of hope, destabilizing her (as in, giving her emotions) and sending her into a quest to find it. After getting beaten by everyone, she gets a pep talk from Mamizou and decides that she doesn't need the mask anymore and will live as she is now.
In 14, the amanojaku Seija Kijin, who was (ostensibly) bullied by stronger youkai, convinced the inchling Shinmyoumaru Sukuna to use the Miracle Mallet (which can only be used by inchlings) to cause a revolution in Gensokyo to make it so that the weak would rule over the strong. Messing around with the Mallet creates a bunch of tsukumogami, including Reimu's purification rod, Marisa's mini-hakkero, and Sakuya's knives, which's what tips the heroines that something's wrong. They beat everyone and things go back to normal (well, Reimu's purification rod still seems to be alive but no one cares it seems).
In 14.3, Seija is being hunted by everyone because there's a price on her head due to her actions in 14. People use impossible spell cards against her, and she uses stolen items to survive. She escapes everyone, and ends up alone and friendless against the world. Which's how she likes.
In 14.5, a bunch of Urban Legends start becoming alive in Gensokyo due to the power of mysterious Occult Balls. Many characters use the power of these balls for their own purposes. Most importantly, Kasen uses it to go to the outside world to search for her arm, and a girl from the outside world, Sumireko Usami, uses it to enter Gensokyo. Nothing is really resolved at the end of this, but Sumireko starts regularly visiting Gensokyo afterwards.
In 15, Lunarian vehicles are spotted in Youkai Mountain purifying the land (ie, utterly destroying everything). Eirin makes a special medicine, the Ultramarine Orb, for the heroines, so that they can properly fight the amaaaaazing Lunarian forces (the medicine makes them sort of see the future, which's why you have infinite lives; the protagonist "sees" how they'd get hit, so they can avoid it properly). The heroines arrive on the moon (using a secret passage through the spirit/dream world), where they discover that the Occult Balls where created by the goddess Sagume Kishin as part of her backup plan to relocate the Moon Capital to Gensokyo (through the materialization of the urban legend that says that NASA discovered a civilization on the moon), which is under attack by a mysterious force who is using fairies (embodiments of life) to bring life to the moon and, as a consequence, death, filling it with impurity (which will make the Lunarians not be ultra special immortals like they like to be). The protagonists then defeats that enemy, Junko, who was attacking the moon because of her grudge against Chang'e, whose husband killed her son. After defeating her, the protagonists also fight her co-conspirator, Hecatia Lapislazuli, who was keeping the Lunarians from waking from their emergency sleep (in which they were put by the higher ups of the Moon so that they wouldn't know about the attack). They defeat her and Junko once more and everything goes back to normal (urban legends are still around because I guess Sagume couldn't be assed to make the Occult Balls stop working)
I wrote all this off the top of my head so some stuff is probably off, but I think the basics are there.
As you can see, Touhou 13.5 has nothing to do with 14; 14 only has to do with 14.3; and 14.5 only has to do with 15. The mangas deal with the urban legends too, but there isn't any ultra relevant info there iirc.