Yes, Satan is alive and well in the world today. Probably more alive today than ever.
Right ! He's often portrayed as being the master of hell, but according to Scripture, he will be just another "citizen" of it. The only special status he will have will be the MOST-HEAVILY-PUNISHED of its inhabitants, as Jesus said it'd be more-tolerable for the land of Sodom than for Capernaum, which heard Jesus preach & saw Him perform miracles, but still rejected Him. Thus, Satan, who often saw Yahweh Himself, but rebelled against Him anyway, will receive the worst punishment of all.
Satan is the source of all sin. All who sin are following Satan, the original sinner, who did so before God made man. The millenium will be relatively sin-free, as Satan will be banished to "the abyss". But meanwhile, he's hard at work now, trying to get as many people as possible to follow him in sin & not come to Jesus for salvation, while also trying to get even the "elect" to sin. He stays busy 24/7 doing this, and, given his might, we need the whole armor of God (faith in Jesus) and His sword (His written word) to resist him. Yes, "resist the devil, & he shall flee from you", but he will be back ! And ONLY GOD'S POWER can resist him!
Please, don't underestimate Satan ! He's the most-powerful of created beings, stronger than the whole creation, & only the HOLY TRINITY is stronger ! But, when you're on GOD'S side, Satan cannot defeat you !