Does the orbital variation in planetary gravity affect the Sun's corona
The gravitational effect of the planets on the Sun can be calculated very accurately. The acceleration of a planet of mass $M$ at a distance from the Sun of $r$ is
$$a = \frac{G M}{r^2}.$$
If you plug in the numbers, you'll find that the largest effect comes from Jupiter due to its enormous mass. The gravitational acceleration of Jupiter on the Sun is $2\times10^{-7}$ m/s$^2$. Even if all planets were aligned so to add up all their gravitational accelerations, the total would only be $2.8\times10^{-7}$ m/s$^2$.

However, the acceleration is not what is important; rather it is the tidal forces, which are proportional to $1/r^3$ instead on $1/r^2$. The tidal acceleration felt over a distance $\Delta r$ at the surface of the Sun is
$$a_{\mathrm{tidal}} \simeq \frac{2GM}{r^3} \Delta r.$$
Now I didn't really bother to read the paper in detail, but the guy doesn't seem to write what exactly in the Sun's atmosphere should be affected by the planets. But even if we say that it's all of Sun's corona, which has a thickness of $\Delta r \sim 10^6$ km, the combined effect of all eight planets standing on a line would be roughly $\mathbf{10^{-9}}$ m/s$\mathbf{^2}$, a minor effect compared to the difference in the Sun's own gravitational field over the same distance, which is 11 orders of magnitude higher.