I put up regular metal gutter, like you would on your house.
And, yes, it is flexible tubing at the bottom. Actually, it was a friend of ours that hangs siding for a living who did it for me, and I was more concerned about the wind ripping it off than anything else, so I asked him to make sure that he added extra screws, or whatever he used to attach it with, so as to keep that from happening.
We didn't have any snow last year, so I hope that isn't a problem.
There is a bit of pitch to that roof, hopefully it melt away and will be okay.
Never thought about snow, at all, actually

.
Usually IF we get snow, it's not much and it doesn't stick around too long.
I had begun to get such a MUCK problem in front of my stall doors that I had to do something. It was becoming a real problem b/c the overhang is only about 1 1/2 foot, and it would run straight down and was washing away my dirt, and with Red coming and going, it was turning into a big hole. YUCK

!
If I had known when I first built the run in, that it would ultimately turn into a fully enclosed barn, I would have designed certain things differently.
For one, I would have had more pitch, and I would have had more overhang also.
But, all in all, BOTH building designs have served their purposes PERFECTLY!
for me, and it worked well with my finances at the time also.
Redboy