You can use the re-run option on a testrail test plan which will then allow you to select which cases to re-run and will add them to a new test plan within the same project.
Some questions on your scenario which might allow better suggestions:
You have 10 projects each having 11 test suites - are they the exact same 11 suites or can they be modified/different for each project? Any reason you can't have a single massive "suite" with 11 sections and only include the appropriate sections for each project/plan?
Are the 10 projects all executed by the same group? It might be easier to have a single project and use plans and milestones to separate the project results if they all share a common set of tests. Your call but cases are copied across projects so updates don't get synced back to equivalent cases in a different project.
Each project has 5 test plans with ~5 suites - does this mean each test plan has ~5 of the previous 11 suites attached to it? What differentiates each test plan within a project i.e. why not just one plan? If they are running the same tests in different configs the test plan configs might be a better way to split the testing.
Hopefully these questions/comments are helpful