if the moon accelerated in the opposite direction as the earth with the same velocity, will it float towards the sun?

I don't think you will ever have to worry about that. Celestial dynamics just doesn't work like that.

No, if it were orbiting the Earth in the opposite direction as it is now, then it would still be in the Earth's gravity well, just going in the opposite direction, which is called retrograde motion. So it makes no difference to gravity which direction it orbits. However, there would be different tidal forces acting on the moon. The Moon, as it is right now, actually steals a little bit of angular momentum from the Earth through the tides. This acts to make the Moon actually start moving a little faster and therefore further away from the Earth. In a few billion years, the Earth and Moon will be both tidally locked to each other, and the Moon will be several hundred thousand km's further away than it is now. If the Moon were in retrograde motion, the tides would work in the opposite direction and the Moon would start moving closer to the Earth not further.

The Moon would have to accelerate to escape velocity or higher, and the direction it would take would depend on WHERE the Moon was in its orbit and what direction opposite to the acceleration is. The Moon could go into orbit around the Sun, become a dwarf planet.

It can not possibly do that, but it probably would

Your question isn't completely clear, but if you mean what would happen if the moon somehow came to rest relative to the sun, then the answer is that the moon would end up in an independent orbit around the sun. It wouldn't accelerate straight towards the sun because of the earth's gravity.

You apparently don't understand physics when you use "accelerate" with "same velocity"