Christians can be oppressed by the devil or demonic spirits, but not possessed. There is no scriptural proof for this.
"When the unclean spirit is gone out of a man, he walketh through dry places, seeking rest, and findeth none. Then he saith, I will return into my house from whence I came out; and when he is come, he findeth it empty, swept, and garnished. Then goeth he, and taketh with himself seven other spirits more wicked than himself, and they enter in and dwell there: and the last state of that man is worse than the first." Matthew 12:43-45
We see from those verses that the unclean spirit was able to come back into the man because the man was empty. Christians most definitely are not empty.
As to animals having spirits, the scriptures say they do.
"Who knoweth the spirit of man that goeth upward, and the spirit of the beast that goeth downward to the earth?" Ecclesiastes 3:21
I can remember a time in my life, when I was 18 (I am 35 now), when I was deeply convicted of sin. I was unsaved at the time, but I read the Bible voraciously. Out of nowhere I had blasphemous thoughts torment me, almost to the point of complete distraction. This went on for about a year. I remember lying in bed one night, and suddenly being paralyzed with fear. I heard what sounded like many people whispering all at once; I couldn't move. It was all I could do to say the name "Jesus" barely above a whisper. When I did, the voices started to laugh, but gradually receded.
It wasn't until I was 30 that I was truly saved. When I was, Satan did still try, on occasion, to harass me. But it wasn't a constant, continual thing.
John, this is my experience. I don't know you personally. It sounds to me like you sincerely want to follow Jesus. I would encourage you to not focus on the devil trying to distract you. Instead, turn your eyes upon Jesus. It almost seems to me that we give Satan more grounds to bother us if we focus on him instead of Jesus, or allow fear into our hearts. Don't be afraid. Remember, "For God hath not given us the spirit of fear; but of power, and of love, and of a sound mind." 2 Timothy 1:7 If this is not your experience, the way is open to you. Jesus died for you so you could be free.