Roter Stern

I'm actually about to (attempt to) recover an old HDD (it's an IDE even), which should have my original CMSF1 install on it.
I ended up losing a number of scenarios and maps I made back in the day, and perhaps this scenario would be among those as well.

Don't do that. This is a mistake that I see a lot by many different people. The Assault command is just a bounding overwatch drill (leap frog). Very useful in many situations, but room clearing is not one of them. For room clearing, you want to have as much firepower directed at the target building as possible, and when you enter the building you do not want to do it piecemeal. 
Here is a very simplified example of how I conduct MOUT:
For simplicity sake, this will only be a 1 squad demonstration using the US Army, but you can scale this up to multiple squads, to platoons, etc, as well as other nations. 
First thing is to split off an Assault Team:

This results in a team of 4 men with rifles and a lot of handheld grenades. The squad leader, a Staff Sergeant, leads the section. This is important as his leadership will help give you an edge in morale in a close firefight:

The second team is the remainder of the squad, 5 men, led by the assistant squad leader, a Sergeant. They have both of the squads SAWs and lots of ammo. This makes them an ideal base of fire element, which is exactly how I am using them in this example:

As you can see, they are on the second floor of a building overlooking the target building, and they have a target arc covering only the target building and the immediate surroundings. This way, they won't get distracted by engaging other targets that appear elsewhere, and will instantly engage any threats that appear in the target building. 
The assault team is given a series of movement points and pauses for its movement to, and entrance into the target building:

First, the assault team will wait 20 seconds before moving into the open. That way, the entire squad has time to orient themselves on the target building and see if there is any initial activity. If you've already had the target building under observation you can skip the first pause.
The first movement order is a Quick move to cover, the low wall. The reason I use a Quick move here is so that if the assault team does take fire from the building, they can return fire as they move to cover. When they get to the wall, they will pause for 10 seconds. This allows them to both catch their breath and observe the building from a closer position. If they were engaged during the run to the wall, this will allow them to put more fire into the threat.

Then, I use a Hunt command on the movement to the outside of the building. This will allow the team to spot better as it approaches, and immediately hit the ground and return fire if they are engaged. When they reach the outside of the target building, they pause again for 10 seconds to look in the windows/doors for any bad guys that might be hiding inside. This pause can be longer, but I recommend at least 10 seconds. 5 tends to be too short.


Finally, the final command is a Quick move order into the target building. This part is very important. It is imperative that once you commit to entering a building, you do it quickly and with the entire assaulting element entering the building. The Quick command is used because the team will run into the building, but will still pause momentarily to engage targets before continuing on. 
 
Others may have a different way of doing this, there is plenty of nuance to MOUT. There are also many situational factors that influence which specific technique is best. I'm using the above as an example of a template you can always fall back on and build from. For example, if you had two squads available you could use one squad as the base of fire element and the second as the assault element. If the target building was occupied and you knew that, you could use the base of fire element to put suppressing fire into the building before the assaulting element moves, and while the assaulting element is moving to the building. 
Do not Hunt into a building. All this will do is get your men stuck in a fatal funnel. The second they take fire, they will immediately stop and go prone. Many times, they will do this in the direct line of fire of the enemy at point blank range, and will get slaughtered in short order.
Avoid rooftops when possible. You are much more exposed on a rooftop, and can be much more easily spotted by the enemy. That said, sometimes the roof provides the best vantage point, or the only vantage point and you have to take a risk. 
All that said, the single best way to clear out a house (or a town for that matter) is to destroy it with indirect or direct fires from heavy weapons, and prod the rubble with a bayonet afterwards, but this is not always possible/practical. 
Hopefully this helps!

Not going to lie, a Russian module for CMSF2 would be a tremendous addition - especially now that the title is released on Steam.
Sure we have CMBS, which can pretty well portray most hypothetical modern day US-v-Russia conflicts, but I think it's save to say that CMBS is now a shelved dead-end product. We are approaching six years since release without as much as even a hit at a single module.
CMBS saltiness aside, I do find CMSF to be more compelling - in both the setting and the variety of OPFOR available. Personally, I'd rather see a 4th module for CMSF than a 1st module for CMBS.
As far as the argument that the Russian appearance in CMSF would "conflict with the lore" - well, that's the beauty of an alternate-history setting - you get the write any narrative to fit your end-goal.
Seems like a fairly reasonable scenario for the Russo-Georgian war to escalate into a Russo-NATO conflict about a year after the events in CMSF. In turn that could bring Russia to Syria in an attempt to re-instate the original Syrian regime, in order to solidify their allies in the Eastern Mediterranean region.
No re-write of the existing CMSF lore, simply a continuation of the original time line past the initial NATO invasion and into NATO occupation. NATO would be portrayed mostly on the defensive and counter-attack; where as Russian forces would be portrayed as a force-multiplier of whatever is left of Syrian military. Perhaps not so much "Syrian military" as much as bands of UNCONS, but that's besides the point.
I would also like to echo some of the things OP is saying (about CMSF:Russia, not so much the extensive apologism of real-life Russian involvement in Syria 🤨):
 

That's exactly my issue with CMBS - all probable additions to the Black Sea theatre have already been done in CMSF; and those not already in CMSF are even less plausible in CMBS.
Sure I get that, I enjoy CMBS a great deal as well - a well equipped OPFOR is a fun challenge; but that's exactly my point.
The way I see it, we can either have CMBS play catch-up to the tune of 3-4 modules ... or get a single new ("well equipped") OPFOR combatant for CMSF and make a solid title even better. Russia is where my vote is, but Iran would indeed make a solid addition as well.
Additionally, playing an interesting BLUFOR, like the Dutch or the Canadians, against a Russianesque and UNCON combatant seems a lot more interesting than commanding Yet Another Stryker Battalion(tm), even if it's against a "real" Russian force.
To be completely honest, I feel the issue of "geographic setting" to be of little consequence. At least to my mind, it is purely an aesthetic issue, one which can be easily solved with a texture pack. To illustrate that point - I have more play-time in CMSF(1) re-textured to look like CMBS, than I do in actual CMBS.
Perhaps that's just me, but I'd rather see a single modern title with a half-dozen modules, than a half-dozen titles with a single module each. 😉

Remember the CMSF2 timeline, 2008. I don't know how much of the fancy-schmancy Russian stuff wasn't fielded yet on that date.
At the time of CMSF1 Steve was adamant about not including 'occupation duty' equipment in the title (I was the guy lobbying to put them in). No Bremmer walls or Hesco barriers, no mine-protection vehicles like RG-31 Nyala (thought the Canadians did get it) or MRAP. He never said it explicitly but I suspect Steve thought it was in bad taste to do an overt Iraq War game while we were still fighting the Iraq War. Maybe now that we're into the third decade of the century Steve would be slightly more amenable to an 'occupation vehicle/force pack'. Thought BFC's go so many irons in the fire currently they're unlikely to get to it.

+1   CM1 was much easier to access in that regards as one could play from Barbarossa to Berlin in one game.  However, the modular method of CM2 development is understandable as it results in more $ for BF's business.  But, development speed of new modules/products seems to be becoming ever more glacial. 
After playing CM2 since 2013 I find that all the WW2 games start to feel the same since they all feature the late war era, and feature more or less the same vehicles and equipment - with small variations of uniforms etc.  If you've been playing these games for a decade, all the WW2 games pretty much play the same/provide the same gameplay experience.  I think we were all hoping to have reached early WW2 (1939 or 1941 at least) by now, and next iteration due soon would have been CM3 with all the improvements we have been listing for over a decade now.
 

Plus 1!