Well, one thing to understand in that ATC in the USA is quite different to Europe.
An airport layout like SFO would be operated quite differently in the U.K.
For example, in the US there's something called LAHSO, which stands for Land And Hold Short Operations. This means that where there are two runways arranged in an X shape, when LAHSO is in progress, an aircraft can land on one leg of the X and be expected to be able to stop before it reaches the intersection, in order to allow independent ops on the other diagonal of the X.
BA 'opts out' of being part of this procedure, as do many European airlines.
On a standard (UK!) cross runway airport, you consider a take off or landing clearance on one to block the other, until that aircraft has passed the intersection. So, if using one runway for departures, the other for arrivals....one aircraft lands, and rolls past the intersection of the two runways. You can then, and only then, clear the departure from the other runway for take off, with the hope (expectation?) that the departure would be through the intersection before you needed to give a landing clearance to the second arriving aircraft on the original runway. Local procedures will detail the arrival spacing needed to achieve a gap big enough to allow the departure.
To answer your question, yes, the departure and arrival routes would be designed to avoid conflictions, or at least most of them.