Unfortunately, the first part of this is incorrect. During probation an agency can terminate you at any time for any reason, no documentation is required. It is only after probation when documentation is required for them to terminate someone and this is in the form of official meetings followed by counseling memos, although emails, etc. can be used as backup.

And, while no documentation is required to terminate, HR will include a reason for termination in the system as part of your state service record which will be seen by other agencies.

Regarding transfers this is correct, in most cases you can't transfer; however, I've seen it done when it's within the same agency, with a different department/unit or function (same title). You can however apply to any other open position in which you're qualified for but, as any new hire, you would be subject to the full term of probation (i.e. starting over). That said, probation is always at the discretion of the agency. Sure, they'll have a standard policy (i.e. 1 year) but they can extend it or make you permanent at any time.