How does Shibata Sensei interact with cards that reduce what you pay, rather than reducing the cost?
Examples: Colonial Dojo School of Wizardry Hiruma Dojo The Allegiance checking trait on Yogo Gingo
Shibata Sensei and Entering Play Cost Reductions
A topic from AEG's L5R CCG Rules forum, copied for Oracle of the Void from a copy of the forum saved in October 2015, so pages it didn't save are missing. Players' names are left out; the rules officials are named. Rulings are as they stood on these dates; later rulings, the rulebooks and the Rules Archive may differ.
How does Shibata Sensei interact with cards that reduce what you pay, rather than reducing the cost?
Examples: Colonial Dojo School of Wizardry Hiruma Dojo The Allegiance checking trait on Yogo Gingo
All of the examples you give either set a stat (gold cost) to a specific value (which in the rules is done by giving it a bonus or penalty equal to the difference between its old and new values) or reduce a stat. Minimums and Maximums are applied after this. Therefore you will still need to pay 5 gold to bring the card into play.