sarangiman, in your case, the extremely close proximity of the ceiling to the top edge of the screen's viewable area is your real deal breaker. The Right sidewall comes in a close second...the Floor a distant 3rd.
As a fact, the HP HC screen is a real start, but consider that even a with a half-gain angle of 15º, when your viewing 16:9 bright content, both the closest surface will still flood the room with light. And while such reflected and returning light will be mitigated by a Retro-Reflective screens performance, especially on which is also a "High Contrast" Gray in shade, when the angle and proximity of such light is so extremely close, the brighter the content, the more pronounced the difference will appear on such a screen. You will see a more pronounced effect along the edges, with the more centralized areas retaining image quality. In order for you to achieve a true elimination of light scatter, your gain on the screen would have to approach 5.0-6.0+ (..and really REALLY be within those numbers....). The big nasty of going that direction is the absolute certainty of seeing graininess and screen artifacts. When one combines high Gain with a Darker surface, the brighter areas within such a screen's surface will always stand out when your viewing at a close enough proximity that you can resolve the individual reflective elements that are needed / used to effect such gain.
No......you need to in at least some manner a way to reduce your surface reflections. If you combine the use of a less reflective covering AND
Retro-reflectivity, then you will have gone the distance required for not just noticeable, but significant improvement.
A within reason solution "screen-wise" that garners at least 2.0 actual gain, combined with a darker but attractive material (...it need not be actually Black...) would be effective enough by my reasoning. 24" out from the Screen wall on the Ceiling and right Sidewall would also be sufficient, far better than naught, but less "in your face" during non-viewing periods.
Be advised that without completely changing your Room's color scheme, you can't get rid of all your reflective bugaboos. But also be advised that by doing at least the minimum described just above, you will effect a greatly improved condition in which to enjoy your content.
Oh yeah...the Floor. Get a Throw Rug, or better still, an appropriately colored Carpet Remnant and have it trimmed/edged to create a 48" x 110" or so cover for the floor below the screen. Would that everything else you need to consider doing be that simple....
Now addressing the Mirrored rear wall.......in reality it will only be an issue in the rear of the room, and even then, between the lessor intensity of the light that has to travel that far to reach the mirrors, and the further attenuation such light will receive as it's collected and reflected off those Mirrors, you can probably reduce their impact by 95% simply by covering them with an attractive, semi-translucent, almost "Gauze-like" Curtain treatment.