What are the forces behind the orbit of the Earth around the Sun?

Yes, it does. Your description is correct. Other bodies than the Sun also pull in Earth with their respective gravities, though, but they are effectively negligible.

The elliptic effect is not as pronounced as you illustrate here, though. Earth's orbit is close to circular.

• When closest, the Sun-Earth distance is around 147 million km.
• When farthest, around 152 million km.

^{From Britannica: Astronomy}

(This image is still not circular enough; see comments. The actual drawn orbit would most likely be impossible to distinguish from a perfectly circular orbit.)

That corresponds to orbital speeds of

• ~$30 \;\mathrm{km/s}$ when fastest and
• ~$29.3 \;\mathrm{km/s}$ when slowest

(according to Ask an Astronomer; I didn't plug in the numbers myself, but that shouldn't be a big deal to do to verify this).