I play a school-owned Holton H378 horn currently, and I wonder if there's an effective way to prevent the noise of the valves. The clicking sound comes from the rotors and out of the bell, and it can interfere with making a good recording. Also, the thumb valve/trigger taps the tuning slide, stopping it from being released completely. How can I fix this?

Check the cork or rubber stoppers on the back of the valve to see if they are worn down to where the valve arm is hitting on metal instead of the stopper. They likely need to be replaced, and this will get rid of the clicking sound.

You can easily adjust the thumb lever so it stops hitting the tuning slide. This will, however, change where your thumb sits and may be uncomfortable at first. Look up a youtube video on how to adjust rotary valve strings or mechanical linkage, depending on what your horn has. If it's a school horn with string linkage, you might want to see if the string needs replaced.

valve re-stringing video:
https://youtu.be/UyxcIWzpxB0

I readjusted my thumb valve and it no longer hits the metal! 👍 But I examined the rubber stoppers. The valve definetly hits the rubber and not metal, but it still makes noise. Could they need replacing even though it's not worn down that much?

Check the stop plate on which the stoppers are mounted. The screws holding it to the valve casing may need to be tightened.

Youtube video:
https://youtu.be/HEPfoHg0m_s?t=25s

If you need new bumpers and your horn has the round slots, I've found that buna-n duro 70, orings are great and cheap. Just find one with the right diameter you need. Slightly more involved, A trick I was shown was to buy brass shim material in .001 and .002 thickness. This cuts very easily. You've gotta know how to take apart the valve https://www.youtube.com/watch?v=q4WIZs6OfFE and if you aren't comfortable doing that, don't. But, basically you cut a very tiny washer-shaped shim that fits the upper thrust bearing and fit it. Usually .001 takes all the upward play out of an instrument, but school instruments may need .002. Out of round bearings also cause issues but you MUST have a swedging tool. It cannot be done any other way. I borrowed a tool from a friend, shimmed, and cleaned my own valves, a 200-300 buck job in about 2 hours and for about 1 buck in brass shim material.

A good oiling can sometimes fix it, rule of thumb: if it moves, oil it! Under the valve caps is the tip spindle on the bottom spindle as well and inside the valves, I also oil the springs and linkage with key oil on my horn

depending how old the horn is it would be necessary to reajust your valves. after some years the valves get more space in the valvestock so you have to go to the instrument workshop pf your joyce. they will tighten the valves inside the cage and grind them in so that there is just as much space they need to move pecisely ( sorry about my english im from germany and i'm missing some of the vocabulary).
ad mentioned above you can also use oil that is a little bit thicker than the usual ones ( hetman is recomended viscosity of 14 ) it works like a small cushion hindering your valves of hitting the sides of your valvecage with too much strength. sometimes the valves geht too much backlash from the grinding of metal against metal