That generator interlock
Don't even think of getting one of those hokey generator interlocks with the 6-10 switches. No reputable electrical switching vendor will make them (ever see an Eaton, Square D, GE or Siemens version? Nope!) At best they're designed for when you can't design the interlock into the system because you did it as an afterthought.
Since this is a blank-sheet design, there are much, much better, simpler ways to do generator interlocks at 1/3 the price, and that let you put any circuit on generator.
The panel size is deceptive, that's 20-space not 40. The "40" claim requires cheater breakers which are illegal on most circuits, as they require full-size AFCI or GFCI.
We'll solve all these problems in a single lick.
The crux is this OEM interlock ($23). It lets us put two Siemens breakers back-to-back ($40 and $10). One is your utility 100A breaker and the other is your generator 60A(?) breaker. These go in the top 4 spaces in a main lug or no-lug panel. Since this steals 4 spaces, we'll bump to this 30-space panel ($66) so we have 26 spaces remaining.
Other vendors make a similar setup, so if you want to shop around, have at.
But this one is Siemens PL, commercial grade -- it even has 2 neutral and 2 ground bars, you don't need to buy them. It doesn't have the bonus breakers so you'll have to replace those. All in all it's about $100 more than your panel choice but it includes a gen interlock that lets you put any circuit on generator instead of just 5. Far easier to wire and there's no question of quality.
Once these 2 top breakers are put in, nothing further need be done with any circuit to make it work on gen - just wire every circuit normally. Even the AFCIs and GFCIs will work properly. When you shut off utility and turn on gen, any circuit you then turn on will feed off gen. Couldn't be easier.
Also, this panel is 125A-ready in case you want to max out your feeder wires.
Other stuff
You didn't mention upsizing the pole breaker to 200A. Feeder lugs there are pointless unless you also upsize the cable to 200A capacity or Aluminum 4/0. "Didn't seem right" - A 200A breaker can't even begin to protect a 1/0 cable let alone a #2. These things can't be left to Providence, that's why we have a Code.
The 1/0 - 1/0 - 2 cable will work fine. #2 is sufficient for 100A under table 310.15(B)(7), and the 1/0 is a bonus hedge against voltage drop. I've been looking for a code cite that allows reducing neutral size, haven't found it, but if it ever turns up, you can kick up to 125A. It's not a service lateral, it's a feeder, and I did find a clause at paragraph 310.15(B)(7) that says you can use the service lateral ratings for feeder that serves the purpose of a lateral.
Conduit
The conduit is a good idea - you'll thank yourself later. Now, you need 18" of earth above the top of the PVC conduit. So if you have to fight for every inch of depth, stop at 21", the pipe is no taller than 3".
Given that you are running in conduit, using cable is wasted and THWN-2 single-conductor is the preferred choice. Cable is legal, it'll just be rather stiff to pull.
I wouldn't glue the PVC either, I assume all outdoor conduit is 100% full of water always, and let the wire's outdoor/wet rating do its job. However watch out for this: Pulling cable will involve a lot of energy, and you could pull the conduit apart in the process. I would leave the ends of the trench unburied so you can quickly fix any pull-aparts. You don't want a pull-apart inaccessible, that will rip up the cable. Ouch.
So here's what we got.
Notice how the utility & gen breakers fit up, easypeasy. After that, I liked your original GFCI MWBC out to the shed, so I put it back in, you don't have to do it obviously. I also show neutral wiring on breakers. I don't show grounds in shed, you know what to do.
Updated with TPeel's main lug feed. If you want the old version hit "edited" just below and look at past edits.
Quality and safety
The cheap Cheese stuff you linked is unspeakably awful, and you really need to stop buying safety related equipment from Amazon Marketplace/eBay/banggood/the other outlets of the Alibaba junkstream. All of those either involve direct ship from China, or dropshipping via the Amazon warehouse which is effectively the same thing.
The Amazon Basics product at least reflects some nominal effort by Amazon to deliver a product which meets its product claims, since Amazon is a bricks and mortar direct seller with boots in the USA, making it easy for consumer protection agencies to hold them accountable for their own stuff. We can't vouch for the insulation quality, but it is unlikely to break down at 24V, because if it did, it would fail its mission as speaker wire.
If you are confused by the difference between Amazon proper and Amazon Marketplace, that's the plan.
Anyway, the real question with this AmazonBasics wire is whether it is a type fit for your use, by which I mean visible vs in-wall, outdoor, etc. The cable rating will tell the tale.
They do make correct cable for such tasks made to go in-wall; thermostat cable comes to mind. Thermostats are 24VAC RMS(about 35V peak) so that's not a problem.
Sizes needed for LED lighting
Low voltage LED lighting is a contradiction. On one hand, you need fairly heavy wire (e.g. 16 AWG) to go any appreciable distance without getting murdered by voltage drop.
On the other hand, LED strips are so delicate that if you solder stiff, fat 18AWG solid wire to their solder pads, the "tail will wag the dog" as it were - the wire will be so much stiffer than the strip that you are likely to tear the solder pads off the PCB. So you are better off pigtailing off the LED strip with #20 or even #22 stranded wire, then come out to a wire-nut to splice to the long-haul cable.
Best Answer
If there are any problems, they'll appear at the terminations first.
So as you work on that system, just keep an eye peeled for any thermal damage at terminations. Discolored wire, melted insulation, arcing, anything like that.
Truth be told, 14AWG wire is allowed for 20A - just look at the table formerly known as NEC 310.16... but this presumes 75C terminations (e.g. the receptacle will not scald your child's finger if he touches the outside while a wire is cooking away at 75C on a screw) and a bunch of other favorable factors. So there is a statutory rule elsewhere in NEC that clamps 14AWG ampacity at 15.