My take, for what it is worth!

Among others,

I believe there was a third [!] assassin who took care of the first two assassins who were probably told to wait by Fredo's bedroom for assistance but were always going to be killed to leave no loose ends

There was a car at the gates trying to drive off or something just after the gun fire and Michael's guards were shouting Stop! Stop! then Halt! Also the sentry was slack leaving the gates open

It seems everyone was pretty much slack after the big party
Rocco failed to secure the compound but “never seen any direct [or even indirect!] evidence” that he was a traitor

If there was the slightest suspicion Rocco was the traitor, I believe there was no way Michael would have let Rocco live let alone continue to be in charge of security and safety of the compound and the family - “trusting [Rocco] with the lives of his wife and his children - the future of the family”

Rocco would have been eliminated the moment Michael was back from Cuba No doubt Michael would “just murder a traitor” Noone gets a pass from Michael not even his own brother

Nobody [including Rocco until he actually saw Michael alive] would have thought Michael survived the shooting
Who can that sort of machine gun fire in such a confined space of a bedroom?!

Rocco's trepidation [we'd try (to take them alive)] in trying to capture assassins armed with machine guns is understandable
How can machine gun armed professional killers be captured alive? Now that is a suicide mission!

Besides even if Michael's assassins were captured alive there would have been so many buffers and still none the wiser as to who gave the order

Rocco's killing of Roth need not have been a suicide mission at all
Rocco seems to have not planned the operation well, no escape plans at all
He could have easily shot Roth, camouflaging his gun and then instead of trying to run away in full view of the Federal agents, with the smoking gun still in his hand! could have just lost himself in the crowd The airport was full of people

The Don might give the killing order to the Capo but it is up to the Capo to work out how to kill the target and get away with it

Neri is deemed to be Michael's Luca Brasi, Michael's personal bodyguard and for special projects [killing of Barzini, the hooker/compromise Geary operation, Fredo] Hence Neri was always by Michael's side

I am not sure about rivalry if any between Neri and Rocco They had different roles and equally important in their respective fields However it seems Neri was more brains than Rocco who was more muscle

If my memory serves me right,
At the “Is it a boy?” scene even though Rocco was already in the room waiting, when Michael, Neri and Tom walked in, Michael asks Neri to get him a wet towel
So I don't think we can read too much into such things – as to who is asked to do what

When Michael said to Neri and Rocco to step outside, Rocco immediately leaves the room

Neri walks over to the table and takes a cigarette or something before walking out Neri also pushes the lock on the door in, before pulling it shut so that the door does not lock presumably for both Neri and Rocco to get back in without Tom having to unlock the door

What I found really interesting was neither Neri nor Rocco engaged in any conversation whilst waiting to get back in. It was like they both avoided and kept their distance from each other!