How the fire spread:
The masonry walls and flying buttresses of a building like Notre Dame support a stone vault, whose underside one sees as a ceiling from inside the building.
Vaults are often substantial structures which are protected from the elements by a separate timber structure that supports the lead or copper waterproofing. This creates an attic space between the top of the vault and the boarding on which the lead is laid. Once a fire is established in such a space, it tends to spread laterally.
The basic structure of these timber structures is normally a series of timber trusses, supporting secondary timber members which span between the trusses.
Some Cathedrals (like York Minster) have timber ceilings or vaults, and if the roof catches fire the ceiling also catches fire and is destroyed. During a fire burning timbers will punch through the vault or ceiling and fall all the way to the floor – and will no longer fuel the blaze at roof level. However, at Notre Dame the stone vault was so strong that the collapsing trusses did not (generally) punch through it. This meant that the collapsed timber members remained on top of the vault, fuelling the fire.
How might the structure be affected by the blaze:
In a structure like Notre Dame the stone masonry walls, flying buttresses and vault act together to create a stable structure which is entirely incombustible.
Historically fires leading to the complete loss of Cathedral roofs were quite common - usually triggered by lightning strikes, and those with stone vaults often survived remarkably well.
The effects on the surviving structure are therefore likely to be secondary: The vault will have had burning timbers sitting on it for some hours. The masonry may have heated up and expanded, which it can do without collapsing. As the masonry cools down and shrinks it may deform.
In addition, much water will have been introduced into a structure that was intended to be protected and kept dry.
The first priority will be to identify which parts of the surviving structure are potentially unstable, and then devise means of providing safe access for engineers to assess how unstable they actually are. From this will come any proposals for temporary propping etc. or careful dismantling of unstable parts.
The next priority will be to assess the risk of problems as the masonry cools down, and devise means of removing the charred timbers sitting on the top of the vaults so that the stone itself can be inspected. That will probably involve work from cranes - so the workers are not standing on something that might give way.
Without the timber roof the masonry will be exposed to the elements, which is not a good thing. Someone will be rapidly working out how to erect temporary roofs, so the process of drying out can start - which is particularly important for the surviving fixtures and fittings.
There is a substantial amount of scaffolding that has survived. That will require its own assessment to see if it is an asset or a liability.