Does She Like Me Back More Than Friends?

SORRY FOR THE LONG DESCRIPTION. PLEASE READ AND I HOPE YOU CAN ANSWER MY QUESTION
SIGNS TO ME SHE LIKES ME
I had asked out a girl who said yes but lied about wanting to go out with me. The girl I like now was surprised that I asked out a girl saying that I have a girlfriend but now, she doesn’t know that the girl I asked out lied. On the same day a boy told me that the girl I asked out was lying about wanting to go out with me. This girl (who I like now) said “Talk to her and find out if she has mutual feeling for you and if she actually wants to go out with you. Just to make sure. She is a nice girl, I don’t think she will be lying.” saying it with a sad face and tone. On the same day, she said she wondered why guys don’t ask her out. (She said this when I was close to her but not directly to me).
The day or two after finding out I asked out a girl, she was talking about our essay project to me and another girl. She read one of the paragraphs and she looked at the girl 3-4 times and to me 7-8 times when reading it. I told her that I thought it was a good paragraph and she was smiling a bit.
Before she found out I asked out a girl she showed signs that to me told me she likes me…
When talking to me about my schoolwork, she touched/tapped my arm and later my back (this has never happened with other girls). Also her hand was close to mine and she didn’t move her hand away and when passing pencils and pens to each other, she always made a way for me to touch her hand or her to touch my hand.
She talks to me and asks questions about my life (If I have brother or sisters) and random questions (If I can do cross eyes, How many children I want when older, about the sport we both like)
She looks at me a lot. She also leans backwards towards me when sitting in class (she sits in front of me).
We were talking in a group and I did something and she said we have to have dinner before we go there and was laughing/smiling
she yelled my name out in a public place (school excursion) when she was with her girl friends and also in class (when in class she yelled my name out a lot throughout the lesson). It was also someones birthday on that same day as the excursion and her friend offered me some cake and she said that she made it.
She also enjoys the sport I like and the girl I asked out doesn’t really talk about the sport. (Seems to have things in common to me)
One of her good girl friends stare/look at me in one of the classes I’m in with her. When I walked passed her group, one of her girl friends looked at me as I walked passed.
I have been having exams and every time I see her after the exam she looks/stares at me. One time she was in her group and I saw her looking at me but I walked passed her. After I saw her with another guy (might has kissed on the cheek) (this guy has a girlfriend) and she looked at me. The next day of exams, she walked passed where I was standing and her eyes seemed to follow where I was standing.
When in class one time, I sat next to this girl and randomly she asked how many kids I would like to have when older. I said how many my partner wants to have but below a certain amount. Without me asking her she just randomly said she wanted a certain amount of children. She was sitting next to one of her girl friends in the class and she didn’t really notice anything happen and the question wasn’t directed to her.
I’m afraid I will look desperate by asking out this new girl. I’m also afraid of asking her out and always say that I like her before asking her out a few days later.
Does she really like me back?
Will I look desperate in this situation?
Is it alright for me to tell her I really like her and then ask her out?