The best way to address the issue of hitting governor limits while testing Apex classes is to use test.startTest() and test.stopTest() methods to reset governor limits. These methods mark the start and end of the test code, and allow you to run your test code with a separate set of governor limits. There is no AppExchange product that can temporarily increase the governor limits, and there is no auto reset property or test.setLimit() and test.
resetLimit() methods that can reset governor limits.
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The architect should recommend backing up the existing metadata using the ANT Migration Tool. To roll back deployment, deploy again to production using backed up metadata. This strategy allows the team to restore the previous state of the production org in case of any issues with the deployment. Creating a sandbox from production to take the backup of existing metadata is not necessary and may take longer time. Using destructivechanges.xml to delete new components is not a reliable way to roll back deployment, as it may not remove all the dependencies and references. Manually deleting new components is also not a feasible option, as it may be error-prone and time-consuming.

Testers will experience functional changes throughout testing due to not having isolation from development.
Using a single developer sandbox for both development and testing is not a good practice, as it does not provide a stable and consistent environment for testing. The developers may make changes to the code or configuration that affect the functionality or behavior of the application, which may cause the testers to encounter unexpected results or errors.

Two software development methodologies that an architect should suggest to solve the issue of visibility are Scrum and Kanban. Scrum is based on the value of openness, which means that the team members and stakeholders share information and feedback regularly and transparently. Kanban is based on the principle of making everything visible, which means that the team uses a visual board to track the progress and status of the work items. Waterfall is not a good methodology for visibility, as it does not allow for frequent communication and feedback. DevOps is not a methodology, but a culture and practice that aims to improve collaboration and delivery across the software development lifecycle.

The sandbox types that the architect should recommend for batch data testing are Partial Copy sandbox and Full sandbox. These sandbox types allow the testing of large data sets that are similar to the production data.
Developer Pro sandbox and Developer sandbox have limited storage and may not be able to accommodate the data needed for the batch data testing.