Effect of Hot/Warm Jupiter atmosphere loss on other planets Let us suppose a solar system with a Hot/Warm Jupiter HJ (with a semi-major axis somewhere from 0.5 AU to 0.0001 AU) and an Earth-Like planet EL in it (with a semi-major axis of about 1 AU). Let their orbital inclinations be a couple of degrees apart. Let the star be a G2V or so, like our Sun.

According to this and other sources, we can tell that Hot Jupiters are "bleeding" atmosphere due to their proximity to their sun. Some of it, I presume, gets pushed out with the solar wind farther out into the solar system, at least as far as the EL is.


What effect will this bled atmosphere from HJ have on EL, if any?
If there is an effect, when does it happen? On all of EL's orbit, despite the different inclinations? Only when EL crosses a transit node? Only when there's a transit (or rather, when EL passes through the transit node a certain amount of time after HJ does, to account for the velocity of the atmospheric particles through space)?
Will this bled atmosphere be (naked-eye) visible from EL? What would it look like?
Would all of this have any effect on life/civilization on the EL?
The primary effects wouldn't be the things you ask about.   First, an object the size of Jupiter would change the day and night sky drastically.    The sky now is basically, Sun, Moon and a bunch of yellow dots (and one reddish dot) mostly fixed, 4 move around visibly to the naked eye (Venus, Mars, Jupiter, Saturn).   

Venus stands out a little as larger and brighter, but it looks pretty much like an abnormally bright star.   All that changes if you put a Jupiter sized planet in the orbits around where Mercury or Venus are.   Jupiter is about 1/10th the diameter of the Sun.   If you put it about midway between Earth and the Sun it would be about 1/5th the diameter and transits over the sun would be obvious, unlike Venus transits which are observable (not a good idea without protective lenses), but easily missed.

Jupiter in an orbit between the Earth and the Sun would have apparent phases like the Moon and visible size variation.  It's likely that in such a scenario, the geocentric model would never have been adapted, cause Jupiter would raise too many questions in this hypothetical.   Aristotle and Ptolomy were smart.   This situation would have been enough to perhaps look at things differently.

There's also the 3 body gravitation problem.   Jupiter inside the Earth's orbit might wreck havoc with the Earth's orbit.  Earth's eccentricity might vary a lot more perhaps leading to more extreme climate change periods.   There's a good chance an Earth like planet's relatively stable orbit would be subject too a lot of undesirable changes with a hot Jupiter between it and the sun.  Comet impacts could be a problem too.   A distant Jupiter changing the orbit of passing comets helps the Earth avoid comet impacts.  A Jupiter that close to the sun could do the opposite.    There's also the migration problem.   If Jupiter forms out far from the sun and migrates inwards, that would wreck havoc with any planet orbits it passes.    So there's a huge gravity problem with your scenario.   If you're looking for Earth like planets, solar-systems with hot Jupiters are probably not the best place to look.


that said, onto your questions.



  What effect will this bled atmosphere from HJ have on EL, if any?
  
  If there is an effect, when does it happen? On all of EL's orbit,
  despite the different inclinations? Only when EL crosses a transit
  node? Only when there's a transit (or rather, when EL passes through
  the transit node a certain amount of time after HJ does, to account
  for the velocity of the atmospheric particles through space)?
  
  Will this bled atmosphere be (naked-eye) visible from EL? What would
  it look like?


It's difficult to say with precision.   Certainly if the bleeding was significant enough it might be visible, but on average I think probably not.   As a bad comparison, we can't see Coronal Mass Ejections.   Most of the "bleeding" from a hot Jupiter would be surface gas molecules, primarily hydrogen and helium, most of it uncharged.  Those gas molecules, spread out in space would be essentially transparent, unless it was an absolutely gonzo insane amount.    Very sensitive cameras might pick them up, but they should be largely invisible from Earth.    

Comet tails are visible, even occasionally visible during the day, but comet tails are different.  They're full of dust that breaks off the comet and the escape velocity off a comet is low, so the dust that breaks off from the comet tends to follow in a similar orbit, hence the relative density and apparent cohesion of a comet tail.   With a hot Jupiter and gas escaping in a variety of directions presumably due to heat escape or Jeans escape primarily, there would be very little cohesion.   Escaped gas molecules would quickly spread out in space.   The charged particles that escape from this theoretical Jupiter might be enough to create a faint Aurora, but should be significantly smaller than the charged particles escaping the sun.   The uncharged particles, a small percentage would collide with Earth, most would miss.   There's no scenario I can imagine that would be visibly impressive from such bleeding unless the hot Jupiter was super-close to the sun, say closer than Mercury and the bleeding was extreme.   Even then, the primary visual would be the stream of gas falling into the sun, not the spread of gas moving towards and past the Earth's orbit.

Something not too different happened in the Earth's distant past.  Venus used to have oceans and it bled probably at least an Atlantic ocean of water, perhaps a Pacific ocean of water into space that drifted mostly past the Earth.  Likely there was very little effect to Earth drifting through a very disburse amount of water-vapor in it's orbit.

That's a ballpark answer anyway.  Corrections are welcome.