Well, how much Tandem ridin have you done? It's a team effort, until you and your stoker gets used to riding together it'll always feel heavy and slow heading uphill. Once you're both accustomed to the bike and each other it'll just feel "different" instead.
And if you have to ride it solo, then Tandems ARE heavy.

Upgrade can mean a couple of different things. Either you can get what's basically more expensive stuff. They might work a bit crisper, but won't really improve on the basic functionality much. Or you can stay in your approximate price range, and simply go for parts better suited for your type of riding.

Unless your current parts are actually worn out or broken:
Derailleur change - won't do you much. Easing up on pedalling when changing gears will improve things a lot more than any derailer change ever can.
Cranks - even less of a change there. Sure you can get those which are stiffer and maybe a tad lighter, which again will mean next to nothing for your ability to tackle a climb.

Pedals - I like SPD-style pedals, but some are intimidated by the much more definite connection to the bike that they offer. In reality you have to be pushing quite hard before there's a measurable benefit from them. I like mine anyhow.

Rear cogs - what do you want an 11 small for? Assuming that you have 700C wheels and a 50-something tooth big chainring, you'd be spinning out at around 60 KMH / 37 MPH. I wouldn't feel any pressing urge to add any more speed in that situation anyhow.

As I can't make out how many speeds you are currently running it gets hard to offer more specific advice. If it's a freewheel hub 7-speed is a so-so option on account of the long unsupported axle. But if you're using 7-speed already, then you could try one of the megarange MTB-style freewheels with a whopping 34T biggest sprocket.

Or you can keep a reasonably densely spaced cassette/freewheel and get yourself a quad crank instead..
Or for the sake of simplicity, just replace your rear right crank with an easily available MTB crank with something like a 24-34-44 or thereabouts tooth count. On 700C wheels it'll still get you to about 50 KMH /31 MPH.

Some people are almost religious about their chains, I'm not. I tend to alternate between Shimano and KMC depending on what I can get easiest/cheapest.
As far as I can tell, running conditions and frequency of clean/lube is more important than the brand.
KMC comes with a quick link, Shimano doesn't.
Chains are almost always listed by the no of gears they're supposed to work with, so just count the number of rear sprockets and you're all set. Length has to be cut to fit anyhow. For your drive chain there are plenty of links on how to get the length right, for your synch chain you'd just have to eyeball it.
IMO using a casing with a liner is more important than the cable.
For an elegant solution you have the extra challenge of finding longer-than-average cables, so you might be a bit limited in your choice there.
I have on occasion successfully lengthened shifter cables using the metal sleeve from an electric terminal block. Simply run both ends through from opposite directions and tighten the screws. It's a bit ugly but real easy.
I've even used butt-spice crimp terminals to extend brake wires. It hasn't killed me yet, but it's not really something I'd recommend. To get it even halfway viable you need a high-end crimping plier, which kinda negates the ease of the operation.
As long as it's clean and not fraying there's not much difference from one cable to another. You're likely to see more change from switching pads and making sure that your routing is as good as possible. Good housing with a liner can make a difference.

change pads, tune the routing. Then you're pretty much out of easy options. You've got an oddball cam-actuated brake which use a non-standard mount, so there's not that much you can do.
One option would be to switch to another fork.
Do note the tandems on good surfaces can put a wicked load on the fork during hard braking, so you might want to get something really sturdy, maybe even custom built. You could even have a disc brake/rim brake combo. I've even seen some dual discs.

Couple of options here. Hook the rear hub brake to a friction/thumb shifter, or hook one of the brakes to a brake lever on the stoker's bar.