Don't have a pic to hand but, if someone else doesn't post one, a chap could sketch the standard thing up quick enough for this discussion ..
Points to recall are
(a) the yawing moment is balanced by rudder input
(b) the rudder input for (a) causes a resulting lateral force imbalance
(c) this force imbalance accelerates the aircraft to the L/R as appropriate (not a huge acceleration but there, nonetheless)
(d) eventually, the aircraft's L/R thrust due to the rudder deflection (causing the velocity component) is balanced by a R/L drag force and we end up back in equilibrium but with two component velocities - ahead and to the side (sidelip velocity)
(e) the two velocity components result in a sideslip angle
(f) to get rid of the unwanted sideslip angle (and reduce the associated drag) one banks slightly into the operating engine side to create a sideslip opposite to that caused by the rudder. Generally, for the main area of critical flight interest (initial OEI climb) the typical bank angle will be in the 2° - 3° range ..
It is for this reason that some jet OEM guidance is not to worry too much about this level of finesse and fly wings level, accepting the minor loss due to slideslip. On the other hand, for a GA twin, the extra performance which one might claw back by banking into the operating engine might just be the difference between going up slightly .. not going anywhere much .. or going down.
Note that, for this discussion, we are talking about performance, not handling - the more commonly known 5° bank OEI relates to Vmc handling and is a maximum to prevent innovatively low Vmc values ending up in the POH. That is, don't fly 5° bank OEI at blue line as you are
(a) wasting your time, and
(b) might just as well fly wings level as the performance for the two situations will be pretty much the same.
a aircraft is then trimmed to fly straight and level
straight infers no sideslip .. level ? ... maybe ... if you a lucky in a light twin ..
All the resultant asymmetric forces are equal and opposite, the aircraft is in balance with no yaw or side slip. The slip ball will be central
Afraid not - if the ball is central ... the wings are level ... and you are in a sideslip and not flying straight. Whether the T/D forces are opposing vectors in the same line is not particularly relevant as that is what the rudder input is about - reject
Thrust and drag are equal but not opposite, the rudder side force is unbalanced, which makes the aircraft side slip towards the failed engine
inconsistent with the original statement (this is OK for the initial transition but not for the final trimmed state) - reject
thrust and drag are equal but not opposite, the rudder side force is countered by a combination of the fuselage side force and the side component of thrust.
that sounds to be not overly unreasonable if we are flying wings level - (but ignores the finesse need for bank to get optimum performance). Reject or accept according to how you want to interpret the question.
Thrust and drag are equal but not opposite, the rudder side force, is unbalanced which makes the aircraft side slip towards the live engine.
nonsense. Once the thing settles down the rudder force will be balanced out by this and that in the way of summed components of force. However, to get a sideslip to the live engine requires insufficient rudder or excessive bank.