Re: Jō Kata
This varies greatly by lineage, association, and even individual teacher.
There have been several shihan that have developed their own systems of aikido weapons training. Most of the list below have passed away, but their systems are still carried on by at least some of their students, sometimes in unexpected lines.
Off the top of my head, in no particular order, with affliations listed for those I can remember: (corrections welcome)
+ SAITO Morihiro (Takemusu Aikido lineages, some still Aikikai affliated)
+ TADA Hiroshi* (ACSA and Aikikai Hombu instructor)
+ CHIBA T. Kazuo (Birankai and USAF, both are Aikikai affliates)
+ KANAI Mitsunari (w/some continued development by students such as Claude Bertiaume*) (USAF - United States Aikido Federation / Aikikai)
+ SAOTOME Mitsugi* (ASU - Aikido Schools of Ueshiba / Aikikai)
+ TOHEI Koichi (possible further development by TOYODA) (Ki Aikido; also includes some instructors that never split from Aikikai, and at least one organization that rejoined Aikikai: Aikido World Alliance)
+ SHIODA Gozo (or possibly one of his students like INOUE) (Yoshinkan/Yoshinkai)
+ HIKITSUCHI Michio (Aikido Shingu)
+ MOCHIZUKI Minoru (possible further development by his son, Hiroo) (Yoseikan)
+ NISHIO Shoji
+ SHIRATA Rinjiro
+ Stephan Stenudd*