Light is horizontal with respect to a mass when it travels between two close points that are the same distance from that mass. This is perfectly normal English usage of the word horizontal. Light from a distant star that just grazes the Sun is initially traveling almost exactly vertically but it is a long way away then, and likewise after it is well past. Only during the closest approach is it horizontal, but this is the part where the bending almost all takes place. Of course it does all the angles between vertical and horizontal at various distances.
I mean anywhere within the body of the Sun and traveling in a direction that is at 90 degrees to the direction of Jupiter at that point. It is normal English usage of the word transverse.

However it is the radiation near the core that is of most interest because the majority of the radiation within the Sun is in the core. Once radiation reaches the convective zone it makes very rapid progress to the surface. Therefore, as a percentage of the mass content, radiation is much more prominent in the core.
Yes. If light traveling transversely to the Sun is bent by the Sun, then light traveling transversely to Jupiter is similarly bent. The amount is just less because of the distances and less because of the lower mass. I have given the proportions that we need to reduce it.
I will give some equations later. However if you don't agree with the logic, then equations will not help. Is there any magic reason why the Sun's gravity should bend light twice as much as Newtonian gravity while Jupiter's should not?
Taken as a system, the radiant energy of the Sun does have rest mass. So yes I mean rest mass in that sense. People are used to taking zero rest mass for photons, but that supposes only that a single photon is traveling in one direction. Once you have a collection then the rest mass of the system is no longer zero but E=mc^2 in the frame in which the total momentum is zero.
Equations are implicit in the proportionalities that I have given. However I will be posting some equations that show how to deal with all the different directions of radiation (radial as well as transverse/tangential). Please be patient. I will be out a lot in the next 2 days but will get it done some time in that interval.
Consider the sun rotating with a planet 6 degrees N of the sun's equator. Then consider the sun rotating day by day over 26 days for a full rotation. Because of the rotation the vector component of that acceleration in the plane of the sun's equator all sum to zero. However the vector component in the N direction is constant. That component is sin(6 degrees) or 1/10 of the total.
Well it works for the one going past the Sun.

However the effects are not really just on the photons because they are interacting with matter more than 10^11 times per second in the solar core. Therefore all momentum that a photon gains from gravitational acceleration is shared with the matter very soon. The important thing that the maths of this will deal with is the time rate of change in momentum per unit mass. In other words acceleration. However we do not use the word acceleration for photons traveling vertically even though there is a time rate of change of momentum. And in this rapid momentum mixing environment vertical photons also are affected and share that momentum with matter. However in the vertical case there is no 2x factor, it is simply 1x the normal acceleration (I will show this soon with equations) which is why I say the effect over all random directions in space for photons in 5/3.
It causes one extra zero. That was answered in the question above. Because the tilt of the orbits relative to the Sun is about 7 degrees. I used sine(6 degrees) which is 1/10 as an approximation.
No at that point I mean Jupiter's orbit around the Sun (and the other gas giants) because it spends 6 years N of the Sun's equator and then 6 years S of it. The other gas giants of course have longer periods.

The orbit around the barycenter is of interest because it shows the sum total of all accelerations on the Sun. Because the photonic content is getting a 5/3 effect in acceleration, it tries to do the whole mass motion about the barycenter multiplied by 5/3. However because it exchanges its momentum with matter about 10^11 times per second, this motion is diluted by matter which must also participate in that motion. If the proportion of mass/energy of photons is 10^-7 of the total then that motion will be diluted 10^7 times.

On top of that the rotation cancels out all of the effects except the N-S motion (which is about 1/10 of the total) so that the final result is that the COM motion is retained only in the N-S direction. However the planetary effects are always only the component out of the Sun's equatorial plane. That means that if Jupiter and Saturn have a conjunction when they are at the far N or far S point it will have a big impact, but if they have it when they are crossing the sun's equator then it will have no effect.

That is why the COM idea almost works but not quite. The period of 171 years will be present on average (it has specific periods of 179 and 159 years between 4 planet events averaging 171 years) as will J-S conjunctions in the same part of the sky every 60 years. J-S have conjunctions every 20 years, and they happen at near to 120 degrees apart in the sky so that every third one is in the same place nearly. The whole thing has a much longer cycle of 880 years as those 3 locations gradually rotate in space until there is a return to the original configuration of J-S.
OK, sorry about that, it is just that they look so nice! :-)