I have read countless articles/threads (including the excellent one from Einsteien) on FD and auto pilot and never came across mention of this behavior. What is the purpose of maintaining a constant roll or pitch when it will eventually put the heli in an attitude sure to bring it down.
While a given stick position may keep the helo level in a hover at zero velocity, a different one will be required to maintain a different flight condition. Helos are by nature unstable in this way; the controls must constantly be adjusted and changed to maintain or change flight direction and velocity. Perfect example of this is the nose wanting to pitch up as you accelerate forward. This is caused by disymmetry of lift across the rotor disks, and countered by applying (and maintaining) more and more forward cyclic as airspeed increases.
If a missile blasts one wing off (it's happened to me before in DCS) you'll have to put in a large amount of lateral cyclic to keep the weight of the other wing from rolling you on your side.
What is the technique that others use to straf in a straight line. When I try to make a quick turn I sometimes find that I have almost no rudder authority so I'll hold in the trimmer to regain it. For those who have strafing down how do you handle the trimmer. I can't imagine your trimming the entire run of the strafe.
The most stable flight condition is going to be forward flight, with the fuselage streamlined through the air. Slow speed hovering or flying sideways while strafing is gonna make you work even harder with those flight control inputs to maintain the pipper on target. Even if a pilot could keep the helo absolutely stable, as soon as you pull the trigger, the recoil from the gun is gonna drag the pipper right, much like if you were to fire an automatic rifle from your right shoulder. Even the helo's flight computer and hold modes can't dampen that.
As for losing your yaw authority, most likely you've trimmed the pedal input to one side, so when you try to apply it in the opposite direction you're effectively moving the pedals back to "center". You can either look down in the virtual cockpit to see where the cyclic and pedals are physically located to maintain reference, or you can enable a graphic that's displayed on the screen to show where the controls are. The command for this is in the Options menu, on the Controls tab, in the section called General ("Show controls indicator").