Hey everyone, I just had 2 quick questions

1.I notice I can hear the noise of my finger fretting the strings, even if I do it very gently. It is more noticeable on new strings, but its frustrating when I'm trying to play something super clean, am I fretting too hard or is this just something I need to ignore?

2. I notice when I play the A form major barre chords on the 5th string it feels more comfortable to play with my thumb close to or over the top of the neck. Also, when I'm playing the A form barre chords I notice a pain going down my hand. When I used to take lessons and play in guitar ensemble I never had this problem, but as I have been trying to get back into these chords after playing a lot of technical death metal (which does not usually have just straight minor or major barre chords) I have seemed to notice this. Do I just need to get back into it or is this a possible medical issue?
1. Do you have your guitar set up properly? Are the strings really high off of the fretboard? Is the neck tension kind of loose? If you answered yes to these questions, that could be why. Are your strings wound? I play with Ernie Ball strings and don't notice noises and such.

What fingers do you hear hitting the string, anyway? The barre finger? Or the other ones? Are you hammering on to the chord, almost? Because you can do that. It's a thing. More information please. Do you use the bottom of your index finger to barre, or do you use the edge on the side of your index finger, where there is less padding? The latter is the easiest way to apply it.

2. Playing with your thumb over the neck is a thing that Hendrix made popular. If you have big hands and can fret notes with your thumb, I'd say go for it. If you can't fret notes with your thumb, screw it because that hand position cuts down your reach by quite a bit when you compare it with the precision you get from having your thumb flat on the back of the neck like a classically trained guitarist.

The pain you notice in your hand is probably caused by improper technique, i.e. wrapping your thumb around the neck to play that chord. I'm not saying that you should never wrap your thumb around the neck -- it has its place when you're bending strings, applying vibrato, or even pulling off to open strings.
My guitar just got a new setup and strings yesterday. The strings are only slightly off the fret board. It mostly happens not during chords but lead playing, shreddy stuff, sweeps etc. I have listened intently and its not open string noise. Its usually rather miniscule, but I notice it when I'm trying to get sweeps and stuff super clean.
Did you set it up yourself or have someone else do it? I always set mine up myself because I don't trust someone else to spend as much time as I would tinkering with it... anyway if you are still getting noise, consider using wound strings (I think they would be advertised as jazz strings?) and also consider if perhaps your action is too low now instead?

If it's not your guitar, it's your muting technique. From what I've seen, most people are good at muting strings below the string they are currently playing with the underside of their index finger, but sometimes people have trouble muting the strings above the one they are playing because they are not using the side of their picking hand thumb to mute but are instead using the soft fleshy part of the heel of their palm or some other soft part of the hand.

Use the side of your picking hand thumb to mute because the side of your finger has that boney ridge that is harder and will mute things faster than something soft like other parts of your hand.

If it's not that I'm not really sure what it could be, maybe your pick? I use Jazz III's and they have a very good attack.