Normally the tank is the wall is the structure. This minimizes weight.
There are usually cylindrical regions connecting the different propellant tanks - sometimes called "intertanks". This portion of the structure does not contain propellant as the rest does.
The shuttle external tank and the Saturn V are examples of that style; I do not know about Falcon but I would be surprised if it is not. There is a blueprint here showing Falcon 1 had integral tankage but I am unsure of the provenance. The Falcon 9 user guide Table 2-1 seems to indicate integral tanks, but the document does not explicitly say this.
I do know of some counterexamples from the past.
One is the old Soviet N-1 rocket, which had spherical propellant tanks inside a fuselage / fairing.
Another design not seen much recently is "cluster" tankage as was used in the Saturn-1B.
Image sources 1, 2, and 3
Edit to address question edit:
Common bulkheads have been used instead of interstages. Two examples are the second stage of the Saturn V and the Centaur-G developed for the shuttle. The Falcon 9 user's guide linked above states that its first stage uses an "insulated common dome".