but where do you place your thumb when your playing? is it supposed to be under the neck or over it?
Imo wherever it feels most comfortable for you.

My thumb tends to move around a lot .... When I do vibrato the thumb is over the neck.
It really depends on what you're playing- mine tends to shift all over. I know a lot of people who use their thumb for chords even.
I keep my thumb over most of the time. The only time I don't is when I'm playing the low strings or when I'm playing barre chords. Occasionally, though, I'll just fret the low string with my thumb.
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I've found that if you want to shred/play fast, keep your thumb straight under the edge of the neck 90% of the time. I say 90% of the time because there are odd times where you have to move turn it a bit. Using your thumb for chords is a bad habit a lot of people have.
it depends on your style of playing. most of the time you will be playing with your thumb on the middle of the neck. some players use there thumb to fret during chords.
the only time I move my thumb from the center of the back of the neck is when I'm sweeping, and even then it's only on 4 or more string sweeps.
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I think most guitar instructors will tell you the correct technique is to place it firmly against the backside of the neck. Obviously, it'll move sometimes, and honestly, I wouldn't worry about it too much unless your thumb starts to get in the way, e.g., by hitting the strings.