I don't know the maths, but is low morale and opposition tactics a factor in allocating shot chance?

No, it isn't. 
Both work towards the GP/DEF/OFF numbers - which in turn define the shot chances.
So they are already priced in, i.e. the shot allocation was already including these effects.

DEE could try a more "balanced" lineup maybe? 
Just 3 defenders listed, and one of them with effective SL of 0.   Any team in the world is losing goals with just 2 defenders.

While I even somewhat agree that DFs have some value, even the worst tactic should roughly provide about the shots it is expected to.
And MSWL isn't a "realistic" football sim, anyway. So I don't quite get the "any team" reference. 
I have oftenly and somewhat successful played lineups with 2 DFs elsewhere.
Counting Chess as a similar battlefield sim, which general would ever attack with his Queen?
 

There are obviously some clarifications necessary:
I don't like losing, but I don't care too much about it, anyway, generally, short time. Shit happens.
Actually I find deterministic games rather boring; I'm not much of a Chess guy.

Still, I only find a certain amount of volatility entertaining.
And the rolls I've experienced were often enough clearly beyond that level.

Therefore I haven't complained about the number of coin flips lost by DEE - simply because it's just normal to lose 50% of your flips.
Even 4 or 5 consecutive flip losses are nothing extraordinary - and way more probable than 1:1000.

If you take my freak suspensions vs RAN: they have roughly turned a ~30% into a ~15% chance, maybe 0.5 expected LP lost.
With 3 suspended players DEE still turned out favourite vs LIV - so the actual effect was mostly negligible.

I was only interested in the frequency of these "freak events", given that I saw way too many in just 2/3s of a season.

I may or may not leave SESL after this season, for the discussed reasons, but I am NOT suspicious of any anti-DEE RNGs.

I've recently grown accustomed to some "interesting" numbers at DEE matches.
I have e.g. already discussed some surprising suspensions recently.

This week's match vs AYR, though, showed up with something special.
DEE had a 9.9% shot chance per minute - getting ONE shot in total.

My understanding of the game engine - please correct me if I'm wrong - was that it rolls 90 times and checks each of this rolls against the shot chance.

So, moderately difficult math:

expected shots = 8.91
prob( shots <= 1 ) = 0.000917 !?

Of course, this was slightly modified during the match.
However, given that this is yet another obscure 1:1000 scenario:
Are those numbers common around here? Or what am I missing?

I don't understand the stats, but if your calculated numbers noted below are correct then by law of averages there should be no more DEE red cards.

Sadly, that's not how the law of large numbers works.