First of all, Hanukkah was established before Christmas. The incident with the Maccabees happened about 100 years before Jesus was born. In fact, Hanukkah, the Feast of Dedication, is mention in the book of John.
I believe the events of Hanukkah happened in the early winter. No one knows for sure in what month Jesus was born. It probably wasn't in winter, since shepherds don't watch their flocks in the open field during winter, Israel's rainy season. Christians probably picked December to celebrate Jesus' birthday to either counteract or Christianize early winter celebrations in pagan societies, such as the "birthday of the sun" during the Winter Solstice around December 21. Therefore, the fact that Hanukkah and Christmas happen around the same time is a coincidence.
(Incidentally, in 2013 Hanukkah will happen closer to Thanksgiving--the same week, I think.)
Now, Passover and Easter happening together is NOT a coincidence. Christians see God's establishment of Passover, when he used Moses to free the Israelites from slavery in Egypt, as a foreshadowing of Christ's sacrifice on the cross, by which he frees those who believe in him from slavery to sin.
In the book of Exodus, in the early spring, God commanded the Israelites to house, then slay, a year-old male lamb (or kid, that is, baby goat), and sprinkle the lamb's blood on their doorposts and lintel. The Israelites stayed inside that night eating the lamb with unleavened bread and bitter herbs. At the same time, God sent the Angel of Death to strike the firstborn of the Egyptians dead in judgment against them. This was the 10th Plague. After this, the Egyptian king, the Pharaoh, let the Israelites go.
Christians see the slain lamb and his blood as the foreshadowing of the slaying of Jesus and the spilling of his blood at his crucifixion. In fact, the Last Supper the night before may have been a Passover seder. Surely Jesus was crucified at Passover time. Then on the third day, Jesus rose. Christians celebrate Easter to commemorate Jesus' resurrection. Holy Communion--the bread and wine as Jesus' body and blood--came out of the Passover meal.
So it is absolutely right that Passover and Easter happen around the same time! Now since Jews and Christians determine their holiday dates differently, sometimes Passover happens about a month after Easter.
But understand that these Jewish holidays were established before the Christian holidays were, just as Judaism was established first. Christianity came from Judaism; Christians believe that Jesus is the Messiah (the Christ) that the Jews were waiting for.