1. So my fortress just hit year 3, and all I've seen are a few goblin snatchers and a goblin thief. Is this just because I picked a peaceful starting area? I was expecting some type of siege by now...
It can vary a lot, especially related to how far you are from the other civilizations that want to take a chunk out you and your population and wealth details (and there's the random factor too). While selecting where to embark, you can use tab to cycle through a list of screens, one of which can verify that you're in a territory that each race can or can't reach (it'll also show if you happen to be at peace, cold war, or outright war with the ones that can reach you, which also has an effect on getting attacked).
And you can also make enemies of your friends, if you wanted to. Abuse a few non-dwarven caravans, that's likely to get a nice war started.
3. I dug out channels that go 3 z-levels down to an arena. How do I safely dump caged goblins and such down there? When I designated the caged goblins to the Pit, the dorfs removed them from the cage and then dragged them to the Pit, all while the goblin was smacking him around. My militia came over and beat him to death once that all started. While it was interesting, I was really hoping to have some !fun! with him in a locked room...
Thieves are 'special' and struggle. Later when you get non-thieves you'll have a much easier time pitting them. The plan to build the cages in the arena and use a lever to open them would work, but is pretty slow to set up, so here's an alternate idea which you might want to try -
This assumes you intend to have friendly units of some type (your military, your warbeasts) killing the gobins in the arena, and not some other sort of torment intended down there.
Stockpile the goblin/other victimtype cages in the arena, with your intended friendly combatants stationed or pastured in there as well.
Place a pen/pasture in this arena (if one's not already there).
Assign your victim to the pen.
A non-military dwarf will release the victim and attempt to move it to the nearby pasture/pen. That handler will attempt to flee as soon as the victim is out of its control - be that from escaping or being placed in the nearby pasture/pen, and the gladiators will move in at this point. If you want this to happen inside a locked arena, just lock it after the handler enters it.