Can an ellipse roll down a tilted sine curve without jumping?

Question

Background

Assume that we have a solid ellipse with uniform density, and that it rolls along a curve.

In the following MO question, I asked along what curve an ellipse rolls down fastest. It was pointed out that one first needs to find any curve the ellipse rolls down from, without losing contact from the curve and jumping up and down. One can consider this as a variation of the classical Brachistochrone problem.

Now, it turns out that there are various curves along which an ellipse can roll horizontally. These are described in this video by Morphocular. It turns out that, if one wants to describe the equation of the curve along which an ellipse with width $2a$ and height $2b$ rolls down horizontally with the center as its axle point, the x-coordinate of this equation comes down to an elliptic integral (see 5:04 of the video):

$$x = \int \frac{b}{\sqrt{1-\epsilon^{2} \cos^{2}(\theta)}} d \theta \label{1}\tag{1} $$

This is not an easy expression to deal with. However, there is a way out if one chooses the foci as the axle points, instead of the center of the ellipse. Assume that the ellipse has width $1$ and height $\sqrt{2}$. In this case, we can describe the curve with a simple sine wave, with period $\pi$ and amplitude $1/2$. Here is a picture (a snapshot from the video at 7:39):

                                              

The equation of this curve can thus be described succinctly as $$y = \frac{1}{2} \sin(2x) .\label{2}\tag{2}$$

Question

Now, suppose that we tilt this sine curve and the ellipse by an angle of $\theta$. In the image below, I have visualized the setting. (Note that the shape of the sine curve is different from the one above. However, this is merely due to my mediocre illustration skills. It is supposed to describe the same curve as the one shown in the image above. The same applies to the ellipse.)

I have the following questions:

1. For what values of $\theta$ does the curve roll down without losing contact with the tilted sine curve - if any?
2. If there are multiple values of $\theta$ for which the ellipse rolls down properly, I wonder: for what value does the ellipse roll down the fastest?