So, as you see, the 'font menu' goes down to the end of the screen. I would like for this menu to show me less fonts, how can I manage this? Thank you. I also don't know what that part with 'EmojiOne' til 'Segoe UI Emoji' is and why it is first (the second one, the first is Recent Fonts), would like to also get that out of there if possible. Thank you.
The fonts you see in the font menu are the fonts installed on your computer. If you want to see less fonts you'll have to uninstall some of them.
I can see that you are on a Windows computer. Here is some info on how to manage fonts on Windows 10.
If you want easier management of your fonts you might try a font manager like the free NexusFont or some paid one like for example FontExplorer X Pro. There are many others. Google "font manager for windows" to find more.
The number of fonts shown in Recent Fonts can be customized in Photoshop under Preferences > Type > Type Options > Number of Recent Fonts to Display. Set it to 0 if you don't want to see any recent fonts:
You might have to close Photoshop and open it again for the change to take effect.
I don't know why those symbol fonts (EmojiOne etc.) are grouped in the beginning of the font menu or how to get rid of them.
To my knowledge, there is no way to make the font menu shorter unless you uninstall some (I wouldn't recommend that). And I can not find anything about what that second grouping of fonts is.
However, if you would like to view a smaller list, or view one that isn't over your working art, Go to Window -> Character and open the Character Palette, you can position that where ever you would like and then open the font list from there.
I keep my commonly used palettes stacked on the right, but workspace is all up to you.