I just did some reading and thinking since I still couldn't figure what my problem is.

But, here it is: I think I fear approaching girls not because of rejection, but because I am afraid of messing up in front of her.

It is not just simply messing up naturally by just saying a wrong thing or something, but it is the sensation of nervousness and fear itself. Basically, when I am in that particular situation to approach as a beginner, I shake and it feels like my head will explode and so on, and this makes me afraid that I might stutter or display my nervousness to a girl.

So I am assuming that this is my primary reason for fearing the approach. I had trouble trying to explain it, so I hope the above was understandable. What do you think?

There is only one cure: go out and sarge. The first times you'll fuck it up (we all did at the beginnings) but soon or later you'll get enough confidence to approach without any fears anyone.
Just a tip: the first times bring a friend with you as a wingman

I agree. I think the term is social momentum. The more you sarge or be social in general, the less you'll worry about messing social interactions up.

_________________
FOREVER A STUDENT OF SEDUCTION

You must be seduced by women in order to seduce women-ElijahGuru

Lloyd- Party All Over Your Body https://www.youtube.com/watch?v=E9DgwZnjI_M

Before the time of a potential approach, what can I do to calm myself down when I see a girl and it is a good opportunity?

Being that my heart rate goes crazy, feels like blood rushes to my head, stomach becomes full with butterflies, my mouth gets uncomfortablele and my arms and legs shake.

what your describing is a fear of rejection, you don't wanna mess up is just another way of saying it. But heres the thing by not approaching you are messing up. If you don't approach you don't get the girl

Just close your eyes, breathe deaply and open her (do this with open eyes!). This relaxation thing shouldnt take more than 2 seconds.

Another way you can get over it is this: you give a friend some money and you are allowed to take it back only if you approach the girl, otherwise your friend can keep the money. One of my friends did this and gave me once to hold 500 euros... And he approached the girl. He fucked it up at a certain point, but he opened her and i gave him the money back. He did this many times and he can now open anyone (without using this little game).
This is just something you could try. It may not work with you, but it could also change your life! Personally, I never needed to give money to anyone for this...