and I will star this by noting that I am NOT a lawyer. I have lived in the DC area for 37+ years and have some understanding of the roles that various law enforcement and military units can play.
The President has complete authority over any Federal land — thus using Military Police in Lafayette Park is completely within his authority.
The mounted police unit that cleared H Street next to Lafayette Park were Park Police, who have some police authority any place in the Metropolitan region, although usually to make an arrest they are supposed to get clearance from at least a Sergeant. I am presuming that the command structure of the Park Police did authorize the use of the mounted unit.
Secret Service have the authority to arrest, to secure, and if necessary to clear areas to ensure the safety of their protectees, obviously including the President.
As far as the flash bangs and the tear gas, given that the demonstration to that point had been completely peaceful it was clearly ordered because Trump wanted a show of force just before he began to make his remarks, knowing it would be covered both on national cable news and possibly on DC local news.
Trump wanted to make a show of going to St John’s Episcopal Church, across Lafayette Square from the White House, which suffered some fire damage yesterday. As far as I know this will be the first time he has been to the “church of the Presidents” although I could have missed a visit. He rarely attends church services, since his Sundays are far more likely to involve trips to a golf course he owns, whether in NJ, VA, or FL (and he usually plays when church services are going on).
As of how Trump has no control over National Guard units. It is not clear he has a legal basis to invoke the Insurrection Act (not that legality ever seems to matter to him or to his AG). That Federal troops were sent to Little Rock by Ike was a very different situation — Gov Faubus using his National Guard to block the entrance of the Little Rock 9 into Central High School was a direct challenge to the authority of Federal courts, so Ike took the Guard away from Faubus and sent in units of Airborne. Rather than regular Army, even military police, it would be more appropriate for him to federalize National Guard units in states with severe disorder where Governors refused to call out the Guard.
Trump admires the likes of strongmen. He wants to be one. The question is whether he will be able to get the regular military to enable his power grab the way he seems to intimate.
We will have to see what happens.