Is there a stable polar solar orbit such that the Earth is always in view and not eclipsed by the Sun?
As ganbustein says, this is not too difficult to imagine. The simplest case (approximating with circular orbits and only the Sun, Earth and Satellite) would have the satellite orbit orthogonally to the Earth with a 1 year orbit. The Satellite will pass the Earth orbit plane in two places, call these "down crossing" and "up crossing" points.
To minimize Earth - Satellite iterations, keep them 90 degrees apart. Have the Satellite over the north solar pole when the Earth is at the "down crossing" point. Then when the satellite gets to the "down crossing point" the Earth will be furthest from the Satellite plane. When the earth is at the "up crossing point" the satellite will under the Suns south pole. And so on.
This would not be completely stable when we include Jupiter and the mutual interactions, but I think they should be small, allowing this to work generally. If someone "does the math" and proves me wrong, I will accept that.
I could imagine a polar orbit that is in a plane which rotates to stay orthogonal to the direction of the Earth. The rotation would be very slow and match the revolution rate of the Earth. The satellite path would be similar to how a ball of yarn is wound.
There's nothing too fancy about this (you could do the same switching Earth and the Sun), so I wouldn't expect any abnormal forces that would decay the orbit quicker than any other. Also, with a little ion engine, it'd be that much easier to maintain.
Sure. Imagine an object orbiting the Sun in a plane normal to the plane of Earth's orbit. Let AB the the intersection of these two planes. (Imagine that A and B are points on the celestial sphere. We're not interested for now in distance from the Sun.)
Then the only time the Sun hides the Earth from the object is when the object is at A and the Earth is at B, or vice versa. The Earth will be at A or B every 6 months, so we only need to arrange that the object is not at the other point at either of these times. Put it in a circular orbit whose period is a multiple of 6 months, and time it so that whenever the Earth is at A or B the object is due north (or south) of the Sun.
Every six months, the Earth will be at A or B, and the object will be north of the Sun and can see Earth. At all other times, the Earth is not at A or B, and the object can see Earth no matter where the object is.