Can this orbit exist around the Earth and the Moon if the weight and speed of the satellite is ideal? I have added a picture below with the orbit drawn.
Objects in that orbit would be subject to significant perturbations from the moon.
Without active maintenance, I can't believe it would be close to that orbit after a couple of years. Most likely it would eventually be ejected into a heliocentric orbit.
You can't put an object into a "perfect" orbit. There's too much other stuff out there (like Mars and Jupiter) that give things slight tugs and make them wander off. In the one you've shown, eventually the moon would toss it aside.