* A supply network cannot draw supplies from another supply network. It can only draw supplies from a naval base which is supplied by convoy from the capital. Ports do not have to be in supply to provide supplies to a convoy but they do have to be connected via land to the capital.
* Units carry supplies with them. As the units advance the local supply
stockpile also advances. Thus units can start a campaign with 30 days of supplies and march through poorly supplied areas until that stockpile runs out.
* HQs consume supplies. Higher HQs could remain further away, and still be in range of the Divisions. As the front advances, then this is less likely, and you need to make a decision between the combat and other advantages provided by the HQs, and the limited supplies. Adding Corps to the hierarchy (which is generally seen as a "good thing") actually adds to your overall supply consumption in an offensive. In some countries, the supply usage of a large force, say an Army Group, may be increased by 10-20% by all of the HQs. It is probably not a good idea for some countries to create Corps HQs as they need to remain close to the front and consume supplies that might otherwise go to the combat Divisions.
* Supply/resources can only travel once across water, not counting straits. Supply will sadly only trace from the depot. Putting more supply in other ports will help supplying those port provinces but little else. It will flood backwards into the system when units move through though. Every non-contiguous to your capital area is consider "overseas" and will have it's own depot. This can be as small as a single hex island or as large an area as several land connected continents. The larger the area overseas, the more difficult it is to supply. An overseas supply depot is treated by the AI just like your capital meaning that all supply to the overseas area will try to flow from the depot to where it is needed.
* While it is correct that the size of the departure port has no bearing on the outbound supply capacity, the infrastructure of the provinces between the capital and that port does affect supply throughput, as well as supply transfer cost for each province you move supplies thru. If at all possible, and not taking into account the vulnerability of a supply line being raided, you want to send supplies either directly from your capital or a hex adjacent to a capital (remember, all hexes adjacent to a capital have MAX throughput). In Italy's case this means manually using Civitavecchia as your outbound port is the optimal choice, as it is adjacent to Rome.
* Supply received via trade is handled exactly like mineral resources. It is delivered directly to your capital, magically skipping across provinces. This happens each day BEFORE supply need is calculated. It is then distributed back into your supply network. Computer programming here, nothing happens simultaneously.
Same for supply created at your factories, it magically skips to your capital, with the following exceptions; If the factory is isolated it stays in the factory hex; and if the capital is isolated it stays in the factory hex. Once that determination is made supply will flow according to supply system demands, barring isolation.
* Another thing that WILL screw up your supply is SRing a bunch of units. While your units may hop a train and move at warp speed to their destination, their supply won't. If you Sr a bunch of units into a region, it will take some time for supply to catch up. You should never ever strat-redeploy in areas where you have low supply and have divisions fighting. The redeploying units will carry off their own supply
pool when they move (even if it means leaving the province dry for other units).
* You can use transport aircraft to dump supplies as close as they can get along a supply route and they'll still be useful. They don't have to be dropped on the troops or an airfield or port for them to get in to the supply system. Anywhere on the route will do. Base them somewhere they have access to a decent amount of supply (supply depot if possible) and select the hex where you want supplies delivered. Watch them, though, as the AI will intercept them if it can.