Mass of an Electron Apparatus
This uses a "Magic Eye Tube" with a circular display that was used to tune radios in the 1930s. My equipment, shown below left, had to be used with an air core solenoid, shown below right. From what I can tell these can still be purchased for about $100 each (solenoid not included) and used as demo equipment.
This video shows the magic eye tube changing as a radio is tuned in and out:
The explanation describes an experiment to determine the mass of an electron approximating the curvature of the "fan" shape using something else round. You seem to need to know multiple voltages, the current and turns in the solenoid (to determine's magnetic field strength) and the curvature of the bend. The instructions end with "Refer to physics text books for the formulae relating to the calculations for the strength of the magnetic field and the velocity of the electrons. From this information you can calculate the approximate mass of a single electron." Not quite spelling it out for us, is it?
I found this image and equations on Hyperphysics that I think match up with this experiment.
I'm not 100% sure that this is the correct equation for finding the mass of an electron with a magic eye tube but it looks promising. The radius of the curve of the electron deflection due to the solenoid's magnetic field would be found by matching it to a circular object. Since it would be hard to accurately measure the radius of only part of a curve, the instructions suggest that if you find something else that has the same curvature but is a complete circle it will be easier to measure the radius. The magnetic field strength of the solenoid could be found by knowing the current through it and the number of coils. The voltage should be measured from the circuit, charge on one electron is known and that should reveal the mass of one electron.
Potential Difference Kit
From the brief assembly instructions I think (1) I'm missing an insulating handle and (2) it is acting like an electrophorus. (Here is an Electrophrus Engineering activity by the way.) That is my guess at least, maybe someone else has a better idea of what this #unknownequipment is used for.
Tangent Galvonometer Kit
The dismantled frame (left) can be assembled (right) to make a base to hold a compass and a loop of wire. A compass (not found in the box) would be placed where the spool of wire is placed on the right. A square loop of wire is made around the four nails (two top, two bottom) that surround the compass resting on the base. This modern version shows the same set-up. There are fancier equivalents of this kit with secured circles of wire around mounted compasses. I've made similar set-ups for students with cardboard in class (start video about 30 seconds in).
I was surprised to read "This can also be used to determine the magnitude and direction of the horizontal component of the Earth's magnetic field." in the description. I found this explanation of the experiment but I'm not sure if this particular rudimentary kit would be successful. This kit is quite large, about a foot tall, and has sharp aluminum edges. This particular one has been defaced with an engraved swear word as well. While it could be useful, I can probably make some that are a bit easier to worth with.
Cathode Ray Tube
I figured it lit up when I saw it but Dan was able to tell me that this piece of #unknownequipment was a Cathode Ray Tube. This Lab Guy post shows how it could be hooked up and made to work like a small TV. That is way above my current summer level of work load though. Dan also said he powers one with a handheld Tesla Coil so until I procure one of those this may not be useful.
While I might be able to get some of these to work, some would require purchase of additional materials. One could find such vintage pieces on eBay but it begs the question, do you want to? They still hold some educational value, if their original purposes are known, but it may not be worth trying to get them to work. For the time being they will remain on a shelf with other vintage pieces I can't bear to part with.