(1) Am I right in thinking that all the Catholic Churches who are in communion with the Bishop of Rome (and this would include Melchites, Maronites, Byzantine Catholics, etc.) are under one "jurisdiction"?
(2)If the above is true, are they all under the same Canon Law?
(I vaguely remember hearing that this was somewhat controversial.)
(3)Do we all use the same Catechism?
(Again, I vaguely recall that, while nobody in the Eastern Catholic Churches would reject anything in the Catechism as an error, nevertheless there are some difficulties because of terminology, style, emphasis.)
With respect to your questions:
1) I don’t think all of the Eastern Churches come under “the direct Jurisdiction” of the Pope. The Eastern Churches govern their own Liturgical, devotional, sprituality matters themselves. Usually, the Bishops and Leaders of Religous Communties among the various Eastern Churches elect their own Metropolitan or Patriarch who serves as the leader of those Respective Eastern Churches. Sometimes perhaps, a list of names may be submitted and the Pope may choose a leader from among a group sent from the respective Eastern Church.
2) There is separate Code of Canon Law for the Eastern Catholic Churches. The Pope appoints a Cardinal to the Roman Curia who is specifically charged with working for the strength and welfare of the Eastern Catholic Churches.
3) There is only 1 Universal Catechesim, from which all Catechisms are based. However, I do believe Pope John Paull II wanted local Churches to prepare Catechesims for various Churches. So, the Maronite Catholic Church can perhaps base their local Catechism on the Universal Catechism, but include more of their own specific Eastern Tradition in it.
However, I think the Universal Catechism did an great job of incorporating both the Western and Eastern Traditions.
Anyway, my stab at your questions. Someone may be able to confirm where I am correct, modify where my answer needs some clarification.
Hope this helps.