When I hit the E string or the A string it makes a weird buzzing noise. It seems like the sound comes from the nut. Maybe it is the sound of the string vibrating in the nut. The A string is easy removable for some reason.
It could also be buzzing on one of the first few frets, meaning you need to raise your action a little bit. Even if the nut was loose (on some guitars it isn't even glued in), the tension of the strings is more than enough to keep it in place and it (most likely) wouldn't buzz. Does the buzzing go away as you fret the string?
it could be the nut yes but it can also be fret buzz. To fix that just raise the action of your strings.
If it sounds like it's coming from the nut then it could be your tuning pegs, if it is then there aint much you can do about it, just buy some new ones
It sounds the same while pressing frets (maybe slightly less sometimes). And I can better describe it as 'rattling'