There are two separate operations mentioned in the OP: Manufacture of the stages that make up the stack, and Integration of the stack prior to launch.
The integration of a 2 stage rocket consists of 2 separate operations: integration of the payload to the second stage, and integration of the second stage to the first stage.
Integration of the first stage to the second stage is traditionally done inside a building. For smaller rockets such as the Falcon 9 and Soyuz, it makes sense to do this inside a standard building. These tend to be too low to stand a rocket up in. You then have the issue of standing the rocket up afterwards. For very large rockets such as the Saturn V and the Space Shuttle integration of the first and second stage is done vertically. This requires an extremely large specialized building and access equipment, but avoids the complications and risks of erecting the complete stack once it is integrated. The larger the rocket is, the more effort will have been made to save weight on the structural design, and the more challenging horizontal integration and subsequent erection will be.
In the case of Starship, the booster is designed to take off vertically, land vertically, and be ready to reuse quickly. The booster needs no heat shield as it only operates at suborbital velocity and most of the parts requiring servicing (in particular the engines) are at the bottom. The booster should never need to be laid down horizontally during normal operations. Hence vertical integration of the upper stage to the booster is the only option. And we know that it will be done outside, not in a building.
To my knowledge we have no information on how the payload will be integrated into the upper stage. It may well be done horizontally for some payloads. The upper stage will require a greater range of maintenance tasks than the booster, and I think it likely that it will need to be laid down horizontally at some point for work on the upper fins and heatshield. After that it would need to be stood up vertically. Here's a video of the space shuttle being stood up vertically and mounted to the external tank. Payloads could be integrated with the orbiter either horizontally or vertically, see section 7.3.3 of the shuttle user guide.
We know from the structural failure of Starship SN3 that SpaceX have cut the structural strength to an absolute minimum, so that for some operations the tanks need to be pressurized in order to withstand the structural load even in the vertical orientation (to say nothing of the horizontal orientation or the transition between the two). So the structural loads for any maintenance or integration operations done in the horizontal will need to be considered carefully. SpaceX are not going to add weight to the Starship for ground maintenance, so any additional bracing will have be be added as external jigs.
Some payloads require vertical integration, and SpaceX wishes to carry these, so vertical integration is definitely going to be available. Payloads requiring vertical integration are often those which have not been designed to take the force of gravity in a lateral direction (in order to save on structural weight) and therefore must be kept vertical at all times.
The most comparable rocket vehicle sections in terms of size and shape to Starship are the Saturn V and the Space Shuttle external tank / SLS tank which were / are manufactured horizonally. These parts are designed to handle both gravity and launch stresses without internal pressure. Starship is not so we can conclude that its tanks will be less rigid. The space shuttle external tank has both longitudinal and circumferential stiffeners. Current Starship prototypes appear to have no circumferential stiffeners, and longitudinal stiffeners only on the bottom skirt.
The individual rings from which starship is made are strong vertically, but if stood horizontally they would deform into ovals, to an extent sufficient to complicate welding of the segments together. Hence SpaceX keep the segments vertical when welding them together to avoid ovalling under gravity. The alternative would be to build a huge jig as long as and as wide as the starship to rotate it and keep the parts true round while welding. You can see from photos inside SpaceX's factory that they have facilities to rotate Falcon rockets horizontally in order to work on them.
Balloon tank design
SpaceX have opted for a lightweight but relatively uncommon structural design concept: the balloon tank. To my knowledge this has only been used before in the examples referenced in the link. The idea is to maintain an internal pressure in the tank to give it rigidity, enabling a reduction in structural weight.
Nominally, Starship is supposed to have sufficient structural strength to handle its own weight when standing on the launch pad, but require internal tank pressure to maintain structural rigidity during launch and flight. Therefore the collapse of Starship SN3 due to loss of pressure in its oxygen (lower) tank during a cryotest in its methane (upper) tank came as a surprise.
I have seen it said that the structure would not have failed if the weight in the methane tank had been that of methane, not nitrogen. While this may be true, it's worth considering what would have happened if the methane tank had been full and Starship had been carrying a payload.
The density ratio of nitrogen/methane is 808kg/m3 / 657kg/m3 = 1.23 . The propellant capacity of Starship is 1200 tons, of which a fifth (assuming ideal fuel / oxygen ratio) is methane. 240 tons of methane x 0.23 = 55.2 tons so the excess mass caused by loading the tank with nitrogen instead of methane was around 55.2 tons.
Therefore we can conclude that if SN3 had been carrying a full load of methane and a payload of 55 tons when the pressure in the oxygen tank was lost, it would have collapsed. It shows just how tight Spacex are cutting the structural design on Starship, and it really cannot take any unanticipated loads. I believe they will either need to redesign the oxygen tank or arrange for the launch structure to provide some support to the upper part of Starship, to guard against structural failure in case of loss of pressure in the tanks on the launch pad.