Is it possible to include information about the status of the need (approved, denied…) and the progress of the training (in progress, not started…) in the Task Plan Needs report?

Yes, this is possible. You can configure these fields in Preferences > Report Configuration > Needs Report Data, for a global configuration for new tasks, but also individually, in each task, in the tab General during task configuration or within the task in the Reports tab. In the tab Reports, select Needs Report Data and then select the fields Need Status and Training Status and drag them to the right column.