I have a number of fields required for the issues. When I try to use Structure's +Next Issue or +Sub-Issue button, the creation of the issue fails, because the values of the required fields were not provided.
You can enter other fields when creating a new issue.
If the initial creation of an issue has failed, you don't have to lose the entered data. Just add the required fields and double-click on the value you need to edit, or click Edit button in the toolbar. You can change the values of the new issue and try to create it again.
For convenience, you can set up a separate view for entering new issues (or modify the preset view called Entry), so you can quickly switch between different sets of columns. See Saving and Sharing Views for details.