This is a kind of thing that is totally counterintuitive until you spend a few (dozen) hours in Kerbal Space Program. Basically keep sails angled at 45 degrees so that the light reflects in the direction you're moving, and you'll be getting closer and closer.

> If you angle the sail to send light in front of you, you will speed up and fall towards the Sun

Don't you mean slow down and fall towards the sun?

It wouldn't be called "the satellite paradox" if it did what you would naively expect. :-)

If you have 2 objects in the same orbit and one accelerates backwards, that one winds up below and in front of the first. This happens because accelerating backwards causes it to lose energy, so it falls, and in falling picks up more kinetic energy than it had before.

It depends on the frame of reference. Obviously we're talking about a solar reference frame here with respect to linear velocity (it's the only other object), but it will also be rotating.

If the reference frame is initially rotating at with your orbit or a closer one, "speed up" is accurate. If it's rotating with some more distant orbit, "slow down" is accurate.

This, by the way, is one reason why physicists tend to just say "accelerate" in both cases: then, it's technically true for any external reference frame.

You will slow down until you are stationary with respect to distance from the sun, and then start accelerating towards it, ie, falling.