172,
The application of flaps is definitely aircraft specific. In airplanes that lose rudder authority with flap application, Vmca increases with application of flaps. This is often the case in aircraft with split flaps, due to the large airflow disruption with flaps extended. Some midwing or high wing aircraft will experience the same thing with flap extention.
If you study out the intent of determining Vmc, you'll find that it's predicated on a takeoff condition, or in other words, the most critical time an engine can fail. At no other time will the aircraft be exposed to as much difficulty and risk as during a takeoff. Hence, power is set for takeoff, flaps are set for takeoff, the aircraft is loaded to the most unfavorable condition. This doesn't at all mean the condition which will produce the highest Vmca...it means the condition most unfavorable for takeoff, and for losing an engien during takeoff (are there any favorable conditions for losing an engine on takeoff...other than being in a simulator??).
When considering published Vmc, many pilots and instructors erroneously assume that it signifies the highest airspeed at which directional control may no longer be maintained during critical engine loss. This isn't so. Vmca may certainly be higher than published Vmc, and it may also be lower.
What does ice do to Vmca? That's the sixty four dollar question, isn't it? The short answer is that it depends. There is no definitive. Icing which reduces rudder authority will mean less rudder is there to counteract assymetrical thrust. Vmca increases. Propeller icing decreases prop efficiency, meaning Vmca decreases. The efficacy and availability of prop deice, boots, etc, all make a difference, as does the symmetry of ice shedding/buildup, and even the type of icing.
As for when to bring up flaps or gear...don't bank on a proceedure because you were taught that way. Bad choice.
Some airplanes are best with the gear coming up first, others flaps first. Some involve flaps part way, and then gear, then the rest of the flaps. Look at the aircraft systems. An open center system with big gear means that only one object is coming up at a time, and the gear may take a long time. It may be a high drag item, while large fowler flaps may be reducing stall speed significantly. Soloution? Gear gets started up early.
In either case, never be in too big a hurry to move anything. An engine failure doesn't require lightening reflexes, and one may do well to follow standard abnormal/emergency protocal when dealing with one. Step one; sit on your hands for ten seconds and think about the situation. Then act; slowly deliberately, and one step at a time. Don't rush. Speed kills. Don't go throwing gear and flap levers; take your time. Get your airspeed, get control, then navigate, then talk, then begin adjusting and moving.
The important thing is to know your airplane. Each airplane is different. Even the same airplane may require a varied technique depending on the phase of flight. A slightly ridiculous example would be caging one on takeoff. The hapless student immediately deploys 50% flaps. The instructor warbles helplessly "What are you doing?!?" The student replies that he understands that as soon as an engine quits, one always goes to 50% flaps.
Of course, the student was meant to retract the flaps to 50%, not add them...but obeying a hard rule without understanding the airpane and the circumstances can be costly.
Consider your systems. Perhaps flaps are electrical, but gear is hydraulic. You just lost an engine, and half or possibly all of your hydraulic capability. What to do? If it's half, your gear may be operating at half speed. This being the case, perhaps you want to get it started up early; it will take longer. You won't be overburdening the system because the flaps are electrical, and these can be brought up simultaneously. Consider your options and work accordingly. It's different for each airplane, and each circumstance.
The most critical thing to know about decreasing Vmca is the most obvious, and it hasn't been mentioned yet...though I know darn well every poster understands it by heart. Pull back the good engine. That will decrease Vmca by all you need. Obviously it also decreases climb performance, and therein lies the crux of minimum control speed single engine flight. This is why gross weight is most unfavorable, even though a lighter weight will result in a higher Vmca.
That emergency landing will be a whole lot easier if it's quiet so you can concentrate, and if you're not spinning uncontrollably. Pulling back the "good" engine accomplishes this quite nicely, should the need arise.