What are the necessary setups for the northeast? I already have a jig rod, a texas rig rod, a topwater rod, a senko/other worm rod, a dropshot rod, and a moving lure rod. What else should I have?

Looks like you are all set! Some like moderate action rods for crankbaits but if you don't mind fishing crankbaits on your moving lure rod I would skip it unless you get into some of the bigger crankbaits.

Is your senko or dropshot rod a spinning rod? can it handle stuff like shakeyhead jigs?

You don't really "need" anything else, but personally I'd get a frog rod.  The lakes I fish have some heavy weeds and pads and I'd be lost without my xx perfect pitch.

I agree, you dont need anything else. You could actually get by with less. IMO a medium heavy fast action rod will do most types of fishing well. And the older i get, I believe it has more to do with the person using the tool (a good quality rod). And specifically if you are a bank beater less gear the better.

As someone who lives and fishes in the Northeast I can say that you have everything covered. The one thing I'd do though it I'd use the T-rig rod as your regular worm rod as well and invest in a shallow cranking rod for smaller cranks. I started using a "finesse" cranking rod a few years ago and my fishing improved, it allowed me to fish small cranks effectively without overpowering them and I got more bites and landed more fish. The reason for this is there are times here when fish will key on smaller baitfish and also a lot of the waters get pressured and the smaller cranks are often the best bait to use but are difficult to fish effectively on a regular medium power cranking rod.

Add the two suggested rods, and then double up...except for maybe the frog rod....so you don't have to retie as often. 

A spinnerbait rod, for windy days,... a frog rod, for summertime weed bed bashing,.... a jerkbait rod, fall is coming and this will really shine,.... a jiggin spoon rod, late fall they will head deep.

I dont know what rods you currently have (specifically per weight limits and such) but these are what I think you will need going by your list.