If screen voltage is much lower than plate voltage (30-40 volts or more) does that contribute to higher cathode current?...screen current?...both?
Half the time, the screen voltage is more like a couple-hundred volts below the plate voltage.
To function as a pentode, a 5-element tube needs to have its screen held at a reasonably constant voltage. Meanwhile, during the input signal swing, the plate voltage is moving both higher and lower, as the plate current works against the plate load impedance (similar to the case of triode preamp tubes and their plate load resistor).
When the input signal swings positive, it lessens bias voltage, cathode current increases, more current has a chance to impact the screen, remaining current goes to the plate, and plate voltage drops (due to plate current, plate load impedance and ohm's law). Peak plate voltage change in an amp with 400v+ on its plates might be near or a little over 300v. If the output tube started with 400v on plate and screen, at peak tube current the screen might be 400v and the plate at 100v. During the negative input swing, the opposite occurs with the screen at 400v and the plate at 700v (all momentarily).
My example assumed screen current didn't increase and cause a drop in screen voltage due to a series screen resistor.
Back to your question: if screen voltage is lowered below some reference point, tube current will be lowered. The relationship to the plate voltage does not matter (unless the plate of an output tube is below maybe 50-70v). That's because that's what makes a pentode a pentode: that plate voltage doesn't impact tube operation, because the "screen grid" is screening the input from the output.
Triodes have different plate curves because their plate current in influenced to a greater degree by their plate voltage. A screen, held at a steady voltage, is what gives pentodes/beam tubes their characteristic plate curves. If you allow the screen voltage to vary (perhaps by connecting the screen to the plate), tube current will vary, and if you connect the screen to the plate the plate voltage will cause plate current changes just like a triode.
If I keep plate dissipation within spec (or slightly over), can I assume that because tube is not red-plating, that the screen current will usually be acceptable too?
Probably. If you start with plate voltage near screen voltage, and plate dissipation stays okay, screen dissipation will
probably be okay.
There are ways you could booger this. But adding a series resistor insures if screen current rises, screen voltage will be slightly lowered, loweringthe screen dissipation.