I think that sounds like a good plan so far, I dont know much about what type of surface covering is best though. Maybe gravel so it can drain through?
Trimming the fence up some and putting some gravel under there in the trench may help move water along to fence line. Does your drain tie into something else?
No I elbowed the drain to drain off into the dirt area in between my driveway and my neighbors driveway. There are hedges that live there so figured was a safe area to drain to. Though we don't get a TON of water throughout the year. But enough that it leaves this area under te garbages muddy.