The field coil is after the rectifier in the power supply, used to drop the voltage as well as filter, but in those days, large value high voltage caps for filtering were difficult to make in a small package but it was really there to remove the ripple.

The additional coil, wired in reverse polarity, is the basis for humbucking instrument pickups and they only came into being in the early 1950s.

WRT filament voltage- the reason those hummed is because that was AC and one or more tubes (or half of a tube) may have been on opposite poles- the hum goes away when they started using DC for this. One of my guitar amps is the same model as the one used when Marshall built his first guitar amp, which is almost a carbon copy but with some components that needed to be changed because the original parts weren't available in Europe- lots of analysis has been done and one site goes into each section, explaining why it does what it does and that particular model of Fender amp can have a very low noise level. It's interesting to read comments from people outside of electronics- they have the opinion that these were designed from the ground up by people who were absolutely brilliant when in reality, most looked in a circuit book to find something for each section that would work for the application. Take a bit of this power supply, use this filter, add this input section, a pinch of tone stack, grab a phase inverter and choose an output section before deciding if they wanted it to be squeaky clean or a bit dirty. Clean (ultralinear) came later, but people are out there, coming up with their own designs all the time.

The designer of the 5F6a Fender Bassman, which is the amp that Marshall copied, was a session musician and if you watched Warner Bros cartoons, you have heard one of the recordings he played on many times- he played steel guitar and the first sound on their theme is the slide 'sweep' that starts low and goes up in pitch.