This application is sending a lot of spam to my phone recently, two or three times a day. I deactivated broadcast messages, tried converting USSD pop-ups into notifications to avoid being interrupted (didn't work). Basically, everything I tried didn't work, so now I want to remove it from my phone. Is it possible to do it without root, with ADB or something? I'm not a power user but I have used ADB to give permissions to some apps before.
If the app in question is stored in /system and it cannot be disabled then it's impossible to remove/disable without root. Android simply doesn't allow "user" to mess with system apps. That being said without knowing device specifics I've seen some cheap chinese phones include adb root which is a somewhat watered down root specific to adb (usually only for debug builds). To test have adb working on a pc and while connected to device and run adb root. If it works it should restart adb service with root perms. But again this is a longshot.
Assuming you get root (one way or another) and you know the name of the app in question find it in system/app or system/priv-app. Mount system as rw and remove the app (I'd recommend copying it to your pc as a backup just in case something breaks).
Here is simple solution to disable SIM Toolkit application. You will need root access for adb. Connect to device with adb shell and run the following commands:
su root pm disable com.android.stk