I like a lot of your selections but I have some personal choices that I would pick-
2000's- While Boston is probably the consensus choice due to the story involved and them doing the improbable and coming back in the LCS, I would have to say personally I preferred the White Sox of this year. They came close to having a historic meltdown, pulled it out in the end, and went on a postseason record of 11-1, sweeping two rounds including the World Series and losing only once, in the ALCS. Plus they had a drought to match the Red Sox's.
1970s- the 1972 Oakland Athletics. They were the flashy team that got the job done when it counted. And come on, any team with the names "Catfish Hunter", "Vida Blue", "Rollie Fingers", etc has gotta make that list.
We already debated the 1890s issue, and I think your choices of either the Braves or Orioles would work fine.