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Unfortunately, it is not possible to restrict the create issues permission based on issue types, neither in next-gen, neither in Classic projects.
We have a feature request opened for about this matter here:
Feel free to vote and watch the suggestion to increase its priority and also receive notifications about any updates.
About the status category, you can change it by moving the columns of the next-gen board:
1 - The most left column is in the To-Do Category
2 - The most right column is in the Done Category
3 - All the columns between both mentioned columns above are in the In Progress Category
Let me know if you have any questions.
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