The -ve grid bias voltage is developed across the cathode resistor, in other words when DC current flows from the cathode to the anode it has to flow through the cathode resistor, that's the resistor that's connected between the cathode and HT -ve (chassis)) the voltage developed across that resistor by the current flowing through it makes the cathode +ve with respect to HT -ve (chassis) the grid is connected to HT -ve (chassis) via a resistor therefore the grid is -ve with respect to the cathode, the grid resistor does not drop any of this bias voltage because under normal operating conditions no grid current flows because the grid is now -ve with respect to the cathode.
The function of a suppressor grid is to repel any electrons that have bounced off the anode (called secondary emission) If the suppressor grid wasn't there those electrons would flow to the screen grid if the screen grid was at a high enough potential, and that, in effect, would shunt the anode load, if the load was a tuned circuit it would reduce it's Q in other words it's selectivity, this was a problem with some hexode mixer valves used in receivers when the screen grid voltage was fed by a single series resistor because when the -ve AGC voltage kicked in the screen voltage would rise due to the reducing screen current and thus attract the electrons that were the result of secondary emission, to help to reduce the screen voltage from increasing with an increasing AGC voltage the screen grid would often be supplied via a potential divider who's total current draw was much greater than the screen grid current and thus had a stabilizing effect on the screen grid voltage. A method of helping to stabilize the mixer screen grid voltage in hexode mixers without a two resistor potential divider was to supply the anode voltage for the oscillator and the screen grid voltage via a common resistor, the oscillator valves internal impedance in effect forming the bottom resistor in a potential divider.
The other downside of the early valves with no suppressor grid was that there was a kink in the Ia/Va characteristic curves, an area of negative resistance if you like where by an increase in anode voltage produced a decrease in anode current and not an increase. Later on kinkless/beam tetrodes were developed that ironed out the kink.
The above are just brief basic explanations which hopefully might help a bit, there are plenty of good articles/books etc on valve basics on the web, on of the best online ones is Inside the Vacuum Tube by J.F. Rider, worth keeping on file:
https://archive.org/stream/InsideTheVacuumTube#page/n1
Lawrence.