"3) The difference in the magnitude of Earth's gravitational field between one end of the tether and the other, helps to orient the tether in a vertical position (known as the tidal force). For a corresponding system in solar orbit, you would surely need a tether some thousands of kilometers in length, at least. But, while difficult to engineer, at least that does seem possible in principle."

Or also called gravity gradient.

As rough measure scale, since Solar tides on Earth are about 1/2 of the lunar tides- this gravity gradient from the Sun should around 1/2 of the moon's at earth distance.

I don't think you would need the tether to be thousands of Km in length, though perhaps this is correct.
But I would guess somewhere between 50 and thousand km.
Also mass is significant factor, so a iron sphere at both ends could reduce length needed. Or perhaps a iron disk at either one. Or one could perhaps employ small solar sail at end furthest from sun.
Also as you say "it helps" but once in a vertical position it should generally speaking stay in vertically position. If one is launching this from Earth and say leaving the earth system via Earth/Sun L-1, one would have the Earth's gravity gradient that one could use in terms deploying it and at L-1 would have both the Earth and Sun's gravity gradient.
A small ion thruster as used in some satellites for station keeping could also be used.