I've done over 30,000 mike on tour with a large Carry Freedom trailer behind my recumbent trike.
Due to my disability I find packing thing down into small volumes hard, so the trailer gives me more room.
I'm around 2 mph slower when pulling the empty trailer due to the add weight and extra aero drag and another 2 mph slower when I add all my camping kit.
As for handling, makes no difference if the trailer is off/on empty/on full apart from the speed reduction.
I've hit 55 mph downhill twice with the full camping load in the trailer and I was rock solid stable at that speed .......

It does take a bit more brakes to stop. But I've twin hydraulic disks up front, so not that much extra.
Two wheel trailers easily flip over on corners if you don’t take care.
Just like my trike, I've never flipped my trailer on a corner. The only times I've flipped the trike/trailer was because of lumps in the road. I'm counting curbs as lumps when you hit them ....

. Hitting a lump hard enough with just one wheel and you can flip the trike/trailer. Running either the trailer empty and/or the tyres at maximum pressure makes it easier to flip a trailer.
I've never used a single wheel trailer due to being on the trike. But I have heard the very odd story from other travelers on tour about a speed shimmy with one. But I don't think thats a common occurance.