Is laying the bike on its side (drivetrain since I have a disc brake) a big deal for oil leakage?

I lay my Rohloff-equipped bike down all the time, including last year on the very route you’re on. While oil leakage is a thing with Rohloffs, in my experience it is a very, very slow appearance of oil on the quick-release and around the external-shifting box, which I periodically wipe off with a soapy cloth once a month or so. It isn’t something that happens at once when the bike is laid down.

In the bikepacking scene, where clickstands never quite took off (perhaps because one is so often around soft soil), all the Rohloff owners are usually laying their bikes down on the non-drive side all the time.

You shouldn't have to lay the bike on the drive side in order to protect your disc rotor, the small oil leakage in my experience doesn’t reach the rotor, and in terms of impact the disc rotor should be protected by your seat stay and rear rack. Laying your bike on the non-drive side is the standard practice, just look at how frame bags are designed to be opened on the drive side.

I'll probably keep a pannier attached on the top side to stop rain from falling into it or find another solution for rain.

The Rohloff is very weather-sealed. You don't need to protect the Rohloff when laying it down outside your tent for the night. If you are still using a chain instead of a belt-drive, though, then of course you have to think about the chain.

Sounds like a great trip.  I nearly always lay the bike down on the drive side, doesn't tend to leak any oil that side.  On a derailleur bike it would be unadvisable, but only because the derailleurs are delicate.  The only thing to consider is that the chain might be picking up a bit more dirt, just wipe and oil it a bit more often.

When Rohloff’s biggest market, Central European commuters, are gradually taking up belt drives, I wonder how much longer Rohloff will continue to produce it’s own chain oil.

It's an oil for chains, rather than specifically an oil for chains on Rohloff bikes, I think it pre dates the hub, back to the time Rohloff were chain manufacturers.  The Oil has a good reputation, I've never used it, I have a preference for biodegradable oils, not particularly for environmental reasons, but because I keep my bike indoors and it's easier to wash off the carpet.

For years Finish Line Ceramic Grease was my all-purpose gunk.

It's optimized for moving parts, but it does well everywhere, for every purpose, starting with assembly. My experience is that the ceramic nano-balls provide enough separation between dissimilar metals to act as a sort of copper grease, at least in my mostly unchallenging use, where my bike rarely gets wet and attracts only enough dust to be wiped clean once or twice a year.

Since I hate getting my hands dirty, it also matters to me that Finish Line Ceramic Grease is clean to work with, and doesn't appear to cause rashes.

Finish Line's Ceramic Grease is definitely recommended by this user.

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I changed my go-to lubricant when Jags (with Danneaux's logistical assistance for international delivery to Ireland) gave me a tube of Phil's gunk, available only in the US, which has the added valuable active characteristic of self-announcing, by color change, when it stops working, for instance when contaminated by water, which is useful for instance in the EXT clickbox on the non-drive axle end of Rohloffs with the external gear change, in which after long experience, I change the grease packing only at the oil change at 5000km or annually, whichever comes first (Rohloff advises repacking at 500km intervals -- total overkill* when you have discovered the right grease), meanwhile contenting myself with a ten second inspection -- 4Gs: grease is green, good to go -- when I think of it, usually around the 2500km mark. It was hell cleaning the Finish Line Ceramic Grease out of the click box when I switched over; it took a couple of tries.

* As always, my advice is aimed at experienced Rohloff users. Newbies and even the less experienced, should stick rigorously to the manual's suggestions until the box is well run-in, which takes thousands of kilometers, because the most valuable thing about the Rohloff isn't the box or even the formal guarantee, it's the goodwill of the Rohloff team when you're in a bind in Outerstan-that-nobody-can-point-to-on-a-map, which you maintain by servicing the box regularly according to the manual.

The Thorn Me'n'U2  is a triplet tandem bicycle designed to carry one adult and two children. It's known for its robust build, these bikes are popular for family cycling and are sought after both new and used. I have had lots of fun with this bike with my kids; they are sadly now too big for it.
In good working order, although the frame has some cosmetic marks.  26” wheels, back rack, good brakes and all the gears you need to get up and down hill. £500 ONO.
Collection only from Oxford.