The Falcon 9 stages, by themselves, are usually stored on rotisserie style mounts, that allow the stage to be rotated and worked on easily. That is, if the engineer needs access to the top, instead of bringing a ladder or scaffolding over, and possibly dropping something on the stage, they roll the stage till the components are at the bottom.
Here is the mount, on a the Orbcomm stage, in the factory, with the engine install rig (The double ring of metal) being pulled back. Look just north of the engines, there is an aluminum ring around the stage.
Here you can see three recovered stages with the ring in site and in various states of engine removal.
Here is another recovered stage, where you can see the rings at both ends.
Here is a shot where you can see the ring from above.
You can see the huge amount of scaffolding they had to deploy for the Shuttle to work in the payload bay, leading to much slower turn around times.
The Saturn V had custom scaffolding in the VAB as well.
With the Falcon Heavy, at some point they have to attach the three cores. Do they attach the cores on the rotisserie stands, or do they wait, bring in the TEL and assemble them together on the TEL itself?
When the boosters are all mated, the second stage is then mated using the rings and the rollers. Afterwards, the entire structure is lifted using 4 spreader beams. (3 bottom, 1 top) 
Each bottom spreader beam has a attachment point for the anchors on the fuselage itself. The upper spreader beam is presumably a strap because there are no anchors there, and it can be assumed from the tank production time-lapse video. 
The entire rocket is lifted, and the TEL comes in below. However, the only part of the Heavy that is assembled on the TEL is the fairing/payload. The payload and fairing goes through a process in the image below. (It is the process for New Glenn, but it is very similar.) 
In another facility, the payload itself is attached to a fixed adapter ring as shown below: 
The fairings itself are attached to a strongback structure with wheels. The strongback structure itself is attached to possibly mounting holes. 
When the entire fairing/payload structure is complete, it is transported onto a module of some sort which gets transported to the HIF (most likely). 
From my own observations, it is highly likely that the same mounting points for the fairing strongbacks are used for the same purpose of lifting the fairing/payload module up. The entire module is then placed onto a structure of some sort already built into the HIF. At this state, the rest of the rocket is already on the TEL. 
The fairing also has its own lower ring which probably has the ability to rotate. Afterwards, the fairing is mated to the second stage.
These statements are solely based off of observations and may not be entirely true.
