When changing the name of a test in the Grade Center using the Edit Test feature, you are editing the Test Canvas in Test/Surveys/Pools, which then carries through to where you have it deployed in your Content menu on the left. It also changes the name of the test in the Respondus Lockdown Browser (RLDB) dashboard in your Course Tools.
If you are not using RLDB, my testing of this shows that you can change the name even after there have been attempts submitted. Though not recommended.
If you have the test set to use RLDB, then yes it will cause a problem. As noted at the very bottom of the RLDB dashboard this will cause a problem even when doing so in just the Test Options of your deployment. "Warning: To preserve the settings used by LockDown Browser, do NOT modify these settings in Test Options: 1) Name 2) Open Test in a New Window, 3) Require a password, 4) Password. If these settings have been modified the test status will appear as an "Error.""
With that said, here is what can and cannot be changed according to the 'More Help' link in a Test Canvas.
The Test Canvas allows you to add and edit questions, add question sets or random blocks, reorder questions, and review the test.
After the test is deployed and students have submitted results, you can edit, delete, reorder, and change the point values of existing questions. Any submissions affected by those changes will be re-graded.
Creating and Editing Questions
New questions can be created by using the Create Question drop-down list.
Questions can also be added to the test from a local file by clicking the Upload Questions button.
The text of each individual question is displayed on the Test Canvas. Linked questions are designated by a globe icon next to the question.
Questions can be edited or deleted by selecting them individually or by using the Question Type drop-down list.
Questions can be edited by selecting Edit from the Contextual Menu. Linked questions are edited in the original test, survey, or pool. Users making changes to linked questions will receive a message warning that these changes will appear everywhere the question appears.
If a test has been deployed and there are submitted results, you cannot add new questions, nor can you change the number of questions presented in a random block, but you can edit, delete, reorder, or change the point values of questions. Any submissions affected by those changes will be regraded.
Changing Point Values
To change the point value of one or more questions, select the check boxes, type a number in the Points box in the header above the question list, and click Update (if the test has submissions, the button is namedUpdate and Regrade). The new point value is applied to all selected questions.
Alternatively, you can change the point value for an individual question by clicking its current point value. A box appears where you can edit the points or designate the question as Extra Credit. Click Submit to make the changes (if the test has submissions, the button is named Submit and Regrade). For tests with submitted attempts, the option to Give Full Credit also appears.
New grades are recalculated for all previously submitted tests and any attempts in progress.
Rubrics
Rubrics can be added to Essay, File Response, and Short Answer questions and used for grading. Questions of these types with associated rubrics display an icon next to the question to indicate the association.
Rubrics may be associated with these question types during creation or editing.
Find Questions
Use this page to search for and select questions to be added to the test.
Create Random Block
Random blocks are groups of questions based on a set of criteria that can be presented in a random fashion determined by the instructor.
Random blocks are groups of questions based on a set of criteria that can be presented in a random fashion, as determined by the instructor. Note that once deployed and students have submitted results for a test that includes a random block, it is not possible to change the number of questions presented.
Create Question Set
Question sets are groups of explicitly chosen questions that can be presented in a random fashion determined by the instructor.
Question Settings
Use Question Settings to set defaults and options for the test questions, including correct/incorrect answer feedback, whether images, files, and web links can be attached to feedback and/or answers, use metadata to organize and categorize questions, set scoring options, and determine how questions are displayed to students.
Ordering Questions
New questions are added to the end of the test. Reorder items by dragging and dropping, or use the keyboard accessible reordering tool.
Note that changing question order only affects new tests (assuming the test is not set to display questions in random order). Attempts already submitted retain the order as originally viewed when the student took the test.
Questions are automatically numbered in the order they are added, and the question numbers will update when items are reordered or randomized. For this reason, use caution when referring to specific question numbers in the question text.
Deleting Questions
You can delete test questions at any time, unless a test attempt is in progress. If there are test attempts, deleting a question removes it from the assessment, along with any possible points earned. Previous test attempts are regraded as if the question had not been included in the assessment.