Great work on the starter. That will make sure you yeast get the job done right.
I'm not sure I would use the combination of the hopped (I'm assuming with English hops) LME and the cascade. You don't typically see these hops together, probably for good reasons. In your recipe above, you have a combination of cascade and English flavor/aroma hops. I would go with only the English.
I wouldn't do a 20 minute boil with the Pilsen DME for fear of DMS. The LME might let you get away with a 20 minute boil, but when you add the Pilsen DME, you probably need to increase the boil time. Even though the Pilsen is DME, it still needs to be boiled sufficiently to drive off the DMS. It would be more of a concern if you were an all grain brewer, but it is still a concern for the extract brewer.
A day or two in the secondary seems like a waste of time and unnecessary oxidation. If you are going 3 weeks in the primary, just go straight to the the bottling vessel. I don't really understand the whole yeast washing and bottling on the same day comment though.
As far as dry hopping, you could go either way since you are doing a hybrid.

For the ESB you don't need it, but for the APA you should have it.
As long as you increase the boil time, the beer should be a pretty tasty hybrid. If you're OK with the characteristics of the two, then rock on.
Let us know how it turns out.