Thanks for your question -- I'll bet many folks probably have asked themselves the same questions.
As we begin to examine the issues you've raised, let's first remember that our discussion of Satan _must_ be limited to what Scripture itself says about our adversary. People across the world talk about "Satan," but in reality, this "Satan" is the figment of their imaginations (mingled with some well-known fables, such as "the Devil has a pitchfork") and not the Satan who is described in God's Word.
We know from Scripture that Satan is a fallen angel (St. Luke 10:18) who once aspired to be on par with Almighty God (Isaiah 14:12-15) and therefore was cast down. He is a liar who attacks God's people and seeks to ravage them spiritually (St. John 8:44; I Peter 5:8). He is real, personal, powerful and persistent in his attacks on God's redeemed (all those who believe on the Lord Jesus Christ as their Savior). We, therefore, are wise to be informed about our enemy and about his evil tactics.
You asked questions about the power of Satan: let's remember that Satan is NOT God. He is NOT omniscient; only the Lord knows our thoughts. He is NOT omnipotent; He brought ill to Job's family _only_ as the Lord God allowed him to do so. Yes, he has a following: his fallen angels, and all those humans who are not believers on the Lord Jesus Christ as their Savior. Jesus tells us in St. John 8:42ff that either we are the children of God (in Christ) _or_ the children of Satan: thus Satan has witting and unwitting (but equally guilty) followers on earth.
Satan has tried to be like God from the beginning, thus he still today tries to persuade people to follow him and to trust that his word is true. His word, of course, is a lie. For example, our adversary calls cheating "fun" and stealing "self-advancement," whereas the one true God calls these sins what they are: sins that merit His wrath.
Surely people "petition" Satan, but their petitions are vain. As we observed earlier, Satan can do only what the Lord allows him to do.
So what about illnesses? Are they the result of Satan's efforts? Certainly not! Pain entered this world because man, being tempted by Satan, rebelled against God and incurred the Lord's righteous curse (Gen. 3:16-19); thus all illness is ultimately the result of original sin. Christians, while free of condemnation from God, will not be free of the curse until we leave this world to be with the Lord (Rev. 21:1). And yes, God allows illnesses to befall us in order to refine us and to make us more like Christ. He alone is both all-powerful _and_ all-loving, and He works out His plan for our lives mysteriously and wonderfully.
In summary, see Satan for who he is: the enemy of Christians, limited in power and in knowledge but anxious to devour the church. And don't remain fixated on Satan: rather, keep your eyes upon Jesus, who utterly vanquished Satan and the power of the grave through his life, death, resurrection and ascension (Col. 2:15; Phil. 2:5-11).