They will have to request a delay of the hearing.
There is case law about depriving an accused of his right to have civilian counsel present, even at an Article 32 hearing.
If they are smart they will all agree to a reasonable delay.
You can file for divorce.
If he contests it then he can exercise his right to try and delay proceedings under the Servicemembers Civil Relief Act. You are required to disclose his military status as part of your filing.
No, a general discharge is NOT automatically changed to an Honorable Discharge. There are people who WRONGLY say that there is an automatic upgrade after six months -- that's not true.
A Soldier discharged with something less than HD or a RE code less than 1 can apply to the Army Discharge Review Board. Here is a link to their website. It is not automatic. The person has up to 15 years from the date of discharge to apply.
If by file you mean make that part of the divorce the answer is yes. VA considers military retirement an asset of the marriage that can be divided at the time of divorce, and from which a spouse can receive a marital share. It sounds like you would be a 20/20/20 spouse which means you could get payments from DFAS once the divorce was finalized. You would keep medical and commissary benefits so long as you did not remarry.
But if you are not divorced and don't have some written separation...
1. You can be placed on legal hold for court-martial prosecution.
2. Depending on how long you have left they can initiate discharge proceedings for an OTH.
3. Orders aren't set in stone. The one exception would be if you have fulfilled your complete contract, not just to EAOS. In which case the only legal hold is for court-martial.
4. If court-martialed or they want to OTH then you should look to contesting this.
You may not have gotten in so much trouble about the adultery. But it appears they may now have some evidence that you lied in an official statement. This is what can be a classic, "it's not the crime but the cover-up" that gets the person into more trouble than might have been if you had exercised your right to silence.
At a minimum I'd expect them to be seeking an Article 15, possibly even a court-martial. You mention "graphic pictures." How were they sent? If sent over a military AKO...
Administrative law would deal with administrative hearings such as those where a medical care provider is being recommended for a loss of licence or limits. But "administrative law" is a generic phrase to cover the practice and procedure before an agency, for example a licencing agency.