For #1, in 3.0.1, you can add "Test cases" Report and "Test Case Review" report into your dashboard, these reports contains the "Test Cases in TestPlan" table, "Test Cases that are Standalone", "Test Plan with its test cases", etc.

There's only one type of Test Case. It doesn't matter if you create a Test Case from the Construction menu or the Add New Test Case button in the Test Case section of the Test Plan. The way you constructed it doesn't make it any different than the other, except when constructed from the Test Cases section of a Test Plan, then the association of the Test Case with the Test Plan will be created for you.

You can see what test cases are in the Test Plan via the Test Cases section of the Test Plan. It list all test cases that are part of that test plan. The Test Cases section of the plan also has an "Add Existing Test Case" icon when you click that, it will show all Test Cases that have not yet been added to the test plan, including any Test Case that was created via the Construction menu.

In reference to your Test Case Execution Record (TCER/TER) question, try going to the Execution->View TER list table, then filtering on the Test Plan name. That way you will see all the TERs that are associated with the Test Plan you are interested in. If you don't see a Test Plan column in the TCER list table, then click the display properties icon on the right side just above the table and add the Test Plan column.

There's also a TER coverage icon in the Test Cases section of the Test Plan, but that only gives a count of how many TERs are associated to the Test Plan for all Test Cases in the plan. It doesn't provide filtering/sorting but it's still a good way to identify Test Cases that don't have any TERs.