Step One: Jamming just any sort of multimeter into a wall socket = NO! Jamming a reasonably proper one into a wall socket is slightly less no, but they still might not like high voltage AC when set on DC. The normal hobby-type multi-meters are not for high voltage or high power, way too dangerous.
To add to it, it means nothing if you do anyway, you may get a reading close to what would be a fair approximation, you may get something way off, nothing is fully guaranteed in DC mode on a normal multimeter. You'll need a true RMS meter to get some real info, or an oscilloscope for good information to act on.
"Humming" in a transformer itself can mean many things and even if you think you can "hear the difference" between all those, chances are beyond good that actually you can't. The majority of the power in your system is AC at the rated frequency, because that is what's supplied. So the transformer will always hum mostly that frequency.
But there can be mains contamination that causes it to hum more loudly, this is most likely caused by one or more devices drawing power with a non-unity power factor or by clunky old-fashioned triac dimmers (especially when dimming "affordable" LED lights). It's likely they are in your own house, but it's not guaranteed. Those contaminations can also be periodical, so they may not introduce any DC effect, but run at 2Hz, 100Hz, 500Hz, 10Hz, many kHz, or just occur balanced at the same periodicity as the mains frequency, but with sharp edges. A (much) higher than mains frequency is less likely to have enough power in a mains line to cause a laminated transformer to make any more sound, but it isn't impossible. Either way, measuring it as DC setting will still give 0V if there is any periodicity balanced around 0 that's "faster" than the meter's update rate, so DC is a very deceptive term to use here.
A power factor is a number that tells you how neatly the voltage and current signal match each other in phase. If they are exactly equal, i.e. the current is 0 when the voltage is 0 and the current is +max at a +max voltage, then the power factor is 1. When they are out of phase (i.e. the current is +max when the voltage is 0) the power factor is 0. While a power factor of -1 would just mean you are supplying energy, i.e. your solar inverter is doing its work: -max current at +max voltage.
Capacitors and inductors in an AC system will cause the voltage and current to go out of phase, because they can store energy, but one will supply when the other takes, so they can compensate each other when you do the maths right.
The rules for power factor are quite strict these days, but used to be slightly slack in "the early days". Power companies don't like it if you have a low power factor in your house, because it means you are using more current than is registered as power in watts, and that current determines how thick they have to make the cables, while watts determine what you pay. Their transformers will become less efficient at low power factors as well.
I would not be surprised if the autotransformer is a partial or possibly full cause of noise in your mains, they are commonly messy. Basically what you do is:
simulate this circuit – Schematic created using CircuitLab
Which means you have an inductance L1, that is over-dimensioned (to allow you to "tap" some power) to move towards transformer effect. The inductance then has an effect on the power factor, but even worse, when loaded it can have all kinds of effects, depending on what your load asks from it.
All that said; overloading also causes excessive humming...
There are two types of transformer (amongst the other types).
You can buy an isolating transformer, the sort that's used for most applications. As the name suggests, these have a mains winding, and a secondary winding, with no electrical connection between the two.
Or you can buy a non-isolating auto transformer. This shares a winding between input and output. While a no-no for applications needing isolation, powering appliances that are designed to be direct connected to the mains is just fine. Because it shares a winding, it's about 50% lighter and smaller (but I doubt as much as 50% cheaper) than an equivalently rated isolating transformer.
What size to get?
If you assume that 350watt rating is the output power of the motor, then it sounds quite reasonable that 1HP (~750 watts) could be the input power.
So 750 watts is a safe size to go for.
However, transformers can be overloaded for a while, and if turned off before they get too hot, that's OK. Does the mixer draw full power all the time, or only when making ground beef? Take those two together, and you may get away with a lower rated transformer that you use intermittently, and keep an eye on the temperature.
Best Answer
Since the amplifier is mains connected you can use an ordinary step down transformer. An isolated stepdown is nice but not needed. The extra safety introduced helps to prevent electrical hazard in case the input or output transformer of the amplifier fails.
If all is good then the common return from the amplifier is not connected to the chassis. The chassis itself can therefore be connected to earth in both situations. Without isolation transformer or standard stepdown transformer.
Finally there is no problem in connecting the amplifier to 50 in stead of 60 Hz mains.
Warning. If you are working on mains operated equipment be careful and respect the fact that a mains voltage can kill.