Does Satan have power to influence us after earth life in Paradise or/and Spirit Prison?
I believe the answer to your question is dependent on what you mean by influence. We read in Alma 40: 11-13,
11 Now, concerning the state of the soul between death and the resurrection—Behold, it has been made known unto me by an angel, that the spirits of all men, as soon as they are departed from this mortal body, yea, the spirits of all men, whether they be good or evil, are taken home to that God who gave them life.
12 And then shall it come to pass, that the spirits of those who are righteous are received into a state of happiness, which is called paradise, a state of rest, a state of peace, where they shall rest from all their troubles and from all care, and sorrow.
13 And then shall it come to pass, that the spirits of the wicked, yea, who are evil—for behold, they have no part nor portion of the Spirit of the Lord; for behold, they chose evil works rather than good; therefore the spirit of the devil did enter into them, and take possession of their house—and these shall be cast out into outer darkness; there shall be weeping, and wailing, and gnashing of teeth, and this because of their own iniquity, being led captive by the will of the devil.
Alma is teaching us that we will, upon death, move from this mortal existence to the Spirit World with all the attributes of who we are as well as who we chose to follow. We need to remember that Satan and his angels never had bodies. Thus, are precluded from the natural processes of this mortal realm. So he will not die and move to the next realm of existence, which is the Spirit World. However, those who have chosen to follow him will undoubtedly leave this mortal life with all the habits, mindsets and burdens that come with living a wicked life (Alma 41:4). Conversely, those who die and have spent their life repenting of their sins and striving to live a Christ-centered and Christ-like life will leave this mortal life and enjoy “a state of rest, a state of peace, where they shall rest from all their troubles and from all care, and sorrow.” They will be restored to be the good people that they have striven in life to become (Alma 41:6).
So while Satan and his angels will not be there, following the path of wickedness will cause a change in our spirit-selves that will continue to influence us in the next world.