If the enemies are engaged doing other things, then I would probably let someone run through, or perhaps make a Dex test to run through if a lot was going on. I'd only have them attacked if there was an unengaged enemy ready for them.

Hmmm, two very different approaches.

We can look to so many different versions of d20 games to see Opportunity Attacks as either the default, impossible/difficult without "held actions" or non-existent because it's also not possible to run past a foe without it being identical to a flee action.

I think I want Opportunity Attacks to be assumed because this group is most familiar with the last three editions of D&D, but also with a reasonably decent chance of pulling off a run-by with a good Dexterity.

So... maybe something like PCs and monsters get OAs by default but one has only to make a Test of Dexterity to avoid any such attack. And one gets a +1 (better chance) of pulling it off for the circumstance of the sideswiper already being engaged. So if the Fighter has already engaged the swine demon, an 18 DEX Thief can run by the demon 6:6 times, but any less agile character is taking a risk.

I think I'll save the Extraordinary Feat of Dexterity for a tumble through ​an opponent to appear on the other side (either to continue running forward or to remain for a backstab).

I think I might also consider a PC or monster who elects to take that Opportunity Attack as hindered (-2AC penalty; p294), since they've perhaps moved their gaze from other potential attackers to get in that extra sideswipe. Not if they've explicitly held for such an opportunity of course. Then the swine demon has an interesting choice about whether to take their eyes off the Fighter or simply let the Thief barrel on towards the orc chieftain behind them.

I'd probably do similar, now that I reread your post -- yeah, a regular DEX test to just run past, and extraordinary test to go through.
I'd be wary of stacking and tracking too many modifiers--I know, that's how I'm used to doing things too as someone who loved 3.x/Pathfinder for 20 years. That's a d20 system misstep though. A good house rule is to just consider all the elements in play--spells, terrain, position, etc--and if after all that the advantage is clearly in favor of one side, just give a single +2 bonus and call it done.

You can't.  You want to get to the M-U in the back of the enemy party, you fight through.  Specifically, if you enter an area where a melee is under way, you stop dead at the edge of the melee, and are now eligible to be attacked.   To me, a melee in AS&SH (and other old-school D&D versions) is a swirling cloud of flesh and steel, not a group of fixed points to navigate through, with or without miniatures. 

Now, if you want to skirt around the edge of the melee, or try a an Extraordinary Test of DX to cartwheel through it (as described in the book), you may have a case with me, but I'll need to be convinced.  My answer will be situation dependent, and that's all you'll do that round. 

I'd also rule that you cannot charge through a melee to get to that sorcerer in the back, as having to to dodge through the combatants it doesn't meet the "reasonably straight path" needed.

I'd have to consider how a mounted rider or a big creature might change my situational rulings.
 

Going through is an Overbear, no? Which is covered and easy to reference and takes a full round. Francisca seems to be in the right of this to me.

Thanks for the feedback everyone. 

When I'm playing Moldvay Basic or similar, I'll be more likely to keep to strict ranks. With its pulp inspirations and all its combat modifiers and options, I feel like this ruleset is looking more to heroic fiction and that warrants for me more of a free-flowing approach to skirmishes than with some of the older games that seem to be traced more directly from the hobby's wargaming roots.

Still not sure what I'll do exactly. Probably just make some situational ruling in the moment and see what people think.