Can I move Checkout issues and data to another project?
The simplest answers is no. The assets and items associated with each other will be broken.
I have been using Checkout in the old format when it existed in it's on project. Now I want to integrate it with other projects, can I move my request types and issues to other projects?
The best cause of action is to move other issues and request types into Checkout project and rename the project. However, you should not rename the project key.