The external dimensions of a 360 Magnum are pretty much identical to a 273 V8's, which could have come stock in your Dart, so the clearance issues aren't going to be a problem.
The motor mounts are similar, though not identical. Schumacher Creative Services in Seattle (their ads can be found in all the Mopar mags) can fix you up with new swap mounts for a decent price.
The 904 Torqueflite, if you want an automatic, shouldn't be a problem as any late '60s, '70s, or non-lockup unit from any L-A engine (318/340/360) will work okay, but the 360 Magnum is an externally-balanced engine, and has a unique spec for the out-of-balance weight. One expeditious way to deal with this is to order a flex-plate from B & M expressly for a 360 Magnum/904 application, and it will come with the correct imbalance; just bolt it in and use a neutral-balanced (318 works great) converter of your choice.
If you choose a manual transmission, other than the out-of-balance flywheel you'll need, I can't think of anything that might be different from a stock 340-size setup. I am unfamiliar with the bell housing sizes with regard to clutch clearance and flywheel sizes for 273/318/340/360 motors, so someone else will have to help you with that. Sorry...
The stickler in this swap is the lack of room for headers, and the lack of any decent cast iron exhaust manifolds that will fit the chassis. The engine compartment on the early (pre-'67) A-bodies is significantly narrower than later models', and things get really crowded down low on the driver's side. There is a lot of mechanical equipment down there (starter, torsion bar, Pitman arm, tie rod, steering link, transmission cooler lines, clutch and shift linkage if it's a "stick", etc.) and not a lot of room for four primary header pipes and a collector.
If money is no object, and you have a week to install them, the TTI system is arguably one of the best, but they are justifiably proud of their products... so, don't be shocked by the tab. They offer complete systems, back to the rear bumper, if you want a bolt-together system that you can be pretty sure will fit with minimal trauma. Let me amend that; "trauma" as in "THIS
&^%%$%$##!!! DOESN'T FIT!!!" They fit, all right, but you practically have to disassemble the front half of the car to get them on. But, most folks who have them seem to think they were worth the fight.
At the other end of the financial scale, there is the Spitfire header, manufactured to cost less... a whole lot less, and still get the exhaust out, probably better than any cast iron manifolds that will fit your car. I say "probably", because I have never seen a flow-bench or dyno comparison; just eyeballed them and made that subjective assumption; could be wrong. Been wrong, before.... more than once!!! Given the alternatives, they MAY be the choice for you. Not my choice, but circumstances alter cases... just don't be in a hurry for them, from what I have heard about delivery dates. Caveat Emptor...
The rear end assemblies in '66 Darts are almost always the 7.25" weak-kneed units that a 360 will be sure to shred if you use its torque potential, so you will probably want to change it out for something stronger, and the options here are many.
The quickest and easiest is probably an original A-Body 8.75" (drop-out third-member-type) from a Duster 340, 340 Dart, or A-body Barracuda, up through about 1972 for the non-E-body cars. These are getting in short supply in junkyards, and a lot of guys are opting for the #2 choice, the later, '73-and up, 8.25" rear ends that replaced the 8.75" units in the Valiants and Darts in 1973. Some Volare/Aspen rear ends are nearly identical in width to the A-Bodys' and have similar specs on the spring pad center-to-center distance, too, but double check those measurements before you buy; I am not at all certain which ones fit.
Another option would be to swap in an 8.75" rear from a late '60s or early 1970s B body. They are wider than an A-Body 8.75", but not by so much that different wheel offset can't take care of that problem, and the spring mounting pads will need to be moved inboard a little; can't remember exactly how much, but I think it's about .75" per side. I've seen people run them by just scrunching the springs over, but that's not a very good idea, I think; stresses the shackles and spring-eye bushings too much. A word of caution; take your tape measure; all B-body rear ends are not the same width.
In most cases, you'll need to have your driveshaft shortened a bit.
The wheels on your Dart, unless they have been altered, have 4"-diameter bolt circles. If you use a B-Body rear, or a '73-up A-Body, or Aspen/Volare 8.25" rear, it will likely have 4-1/2" bolt circles, and you'll be carrying two spares, unless you swap to '73-up disks in front, which will give you the large bot-circle all around.
The 8.25" rear is pretty strong, but the 8.75" is stronger, yet, so, if you plan to do a lot of drag racing with slicks, it might be the better choice.
There are several other issues to deal with, such as the induction system. Do you want to run the Magnum fuel injection system, or swap to a carburetor? There are lots of intake manifold choices for single 4-bbl systems that fit a 318/360 Magnum. If you run a carburetor, you can use a conventional mechanical fuel pump by changing the front cover to an early model, which has a boss on which to mount a mechanical pump (and, the attendant hole) and purchasing an add-on fuel pump eccentric from Hughesengines.com, a Mopar cam grinder in Illinois. It's an off-center, circular piece that bolts to the end of the Magnum cam and, it activates the fuel pump arm. I have run one for years without incident. Keyword is "LOC-TITE" LOL!
Expect a BIG increase in gas consumption, and acceleration...
My single 4bbl, 360 Magnum with a VERY mild cam and stock 340 exhaust manifolds hauled my considerable a$$ through the quarter mile in a relatively heavy '72 Valiant in 13.35 seconds on a 95 degree day, through the mufflers... with an "open" 3.91 gear.
Cheap power, if you don't get greedy... Your car will be quicker and faster than mine because it's probably 200-300 pounds lighter. The early models were more "svelte.".... LOL!
I 've had nothing but fun with my 360 Magnum/A-Body... I recommend it; lots cheaper than a therapist... and, more smile-inspiring!
Good luck with your Dart!!!
Bill, in Conway, Arkansas