How can I lower the volume of the sounds of mastery emotes and level ups? At the same time I don't want to change the volume of spell effects and other meaningful sounds. So far, I've tried all the options in the sound settings. Do I really have to lower the master volume and up everything under it?
There is no option to lower those two sound effects specifically. I suspect those belong in the "Sound Effects" category, so the only way to lower those volumes is to also lower the sound effect volume, which you can do in-game by opening the menu and looking the audio options.
You do have another option for the mastery emotes though. If there is a player who is spamming their emote and you find it annoying, you can mute that player by typing
ignore PlayerName or by clicking the mute button next to their champion portrait on the stats window. This will prevent you from seeing/hearing their mastery emote, as well as their other emotes as well (dance, laugh, taunt, etc). The downside is that you will unable to see anything they type in chat... but then again, if they're spamming their emotes they probably don't have anything too useful to say anyways.