We always run 1.5" pvc with electrical sweeps to the outside of the decking. Then you seal the pipe in the pot. You then run your drip line up into the pot through the pipe, and water can drain back out and not make a mess all over the travertine.

There should be a swale carved into the yard to direct drainage away from, or around, the buildings and out to the street or other drainage. Here's a picture showing what I mean. You can see how the contour of the soil keeps the runoff from flowing down to the lower yard.

I'd also suggest trimming back the step treads to just a 1" overhang past the outermost stringers. That huge overhang you have is why the wood is already starting to twist and the end of each board is pointing a different direction. Do a handrail right at the house corner and you'll be a lot safer. Stupid question - when you say you intended to create a downspout, does that mean you have a gutter but no downspout? So water is shooting out the gutter to land in the barrel? If I'm understanding that correctly, you need a downspout. Downspout goes into rain barrel, there's an overflow pipe at the top of the barrel, and that is piped away from the foundation. If you have that volume of water just dumping there from on high, that's a huge part of the problem

A surface drain or (even dug lower) a french drain will not help if you don't have the slope to daylight the drains. And even IF the city allowed you to cross under the sidewalk with a pipe, you can see from the picture that you do not have the slope, nor curb style, to daylight a drain to the street. Assuming that the city follow ups ok'd the distance the neighbor's sump discharge is away from the property line, and lacking any place on the neighbor's lot for a retention area of the sump discharge, the best scenario (without pics) seems to be to have the neighbor route their discharge farther away from your lot, like into their main front yard so there is more field to accept the water. And if you edge the sod to the concrete (leave at least 1/2" gap) then it will help with some water to not flow onto the concrete surfaces.