Here is a link I found online to a typical Executive Branch agency’s rules on Official Records creation, handling, retention and disposal. In this case it is USGS. One would believe that similar rules would apply to the Office of the President itself.
A typical agency manual on Official Records
From reading the above linked document one can determine that Email as created as part of one’s official duties is considered a record. Improper creation or handling of official records is illegal as defined by the statutes.
Please read more below:
Here's an excerpt from the document linked above:
Official Records Defined. 44 U.S.C. Chapter 33, Basic Laws and Authorities of the National Archives and Records Administration (NARA), defines records as "...all books, papers, maps, photographs, machine-readable materials or other documentary materials, regardless of physical form or characteristics, made or received by an agency of the United States Government under Federal law or in connection with the transaction of public business and preserved or appropriate for preservation by that agency or its legitimate successor as evidence of the organization, functions, policies, decisions, procedures, operations, or other activities of the Government or because of the informational value of data in them."
And more:
Problem Areas. The NARA has found that documentation practices are often weakest in the areas of policy and decisionmaking, particularly oral decisions and commitments; board, committee, and important staff meetings; drafts and working papers or files; electronic information systems; and contractor records. The value of these types of files dictates the need for special attention in those areas.
Documentation of Policy and Decisionmaking. All information supporting agency policy and/or decisionmaking, regardless of form, is considered to be part of the records of that decision or policy. Record materials include documentation of phone conversations, facsimile copies, electronic mail (e-mail) printouts, meeting minutes, and textual records and should all be kept along with the final record copy of any policy or decisionmaking document.
Here’s the kicker:
Records Retention. Records, either permanent or temporary files and including information stored in any automated storage media, include those documents that the agency creates or receives. The destruction or other disposition of Government records is prohibited by law unless authorized by the Archivist of the United States.
This is just a sample, there’s a lot more detail in the entire document as linked above, including references to the statues that apply.
I am not a legal professional, but as a layman I believe that maintaining an email stream of "official government business" on non-government servers is a huge no-no and breaks laws.
Further reading below from the National Archives:
National Archives and Records Administration (NARA) statutes:
An excerpt from this link:
§ 3106. Unlawful removal, destruction of records
The head of each Federal agency shall notify the Archivist of any actual, impending, or threatened unlawful removal, defacing, alteration, or destruction of records in the custody of the agency of which he is the head that shall come to his attention and with the assistance of the Archivist shall initiate action through the Attorney General for the recovery of records he knows or has reason to believe have been unlawfully removed from his agency, or from another Federal agency whose records have been transferred to his legal custody. In any case in which the head of the agency does not initiate an action for such recovery or other redress within a reasonable period of time after being notified of any such unlawful action, the Archivist shall request the Attorney General to initiate such an action, and shall notify the Congress when such a request has been made.
I hope some Attorney-type Kossacks are able to look at these references to Official Records statues and gain some understanding of what the law says and whether or not laws have been broken.