To answer your question: Water will flow through a pipe at any slope. (Pour a glass of water on a tabletop to see what I mean.) What you want is to keep water flowing at a decent rate to move sediment rather than letting it accumulate. Shallower slopes are fine when the pipe is larger, but I’d try to keep to plumbing standards. So, a minimum of 1/8"/foot (1%) to 1/4"/foot (2%) will keep water moving along at a decent rate. Tie a string to stakes, adjust it with a level, and do the math based on the drop over the length.
However, I think that there are some other issues raised by your post that may not have been addressed.
It appears that you are collecting storm runoff from your roof by means of gutters and downspouts, directing them to a common drain, and then having that drain outlet through a retaining wall adjacent to a sidewalk in a public right-of-way. Did I get that correctly from your photos?
If so, just bear in mind that most storm drainage regulations require a property owner to control that storm drainage on the same property, and not to direct it to another property (either public or private). There is some flexibility to this, of course. For example, regulating authorities tend to turn a blind eye toward homeowners while holding commercial property owners to meet this requirement. In older cities (especially those with small building lots) before people paid much attention to storm drainage control, there is a lot of pre-existing flow from private properties to public ways, and it is difficult or impossible to keep this runoff on your own property. However, in your case you are taking drainage that currently flows over your yard, spreading the flow out and allowing it to infiltrate a bit, and you are concentrating it to one point. Imagine walking down the sidewalk by that outlet, and you can see that a pedestrian might as well be walking in the gutter. It is entirely possible that no one will object to your design solution, but be prepared if someone comes knocking on your door.
Let’s assume that your solution is acceptable. You are not very deep at all with your drain. So don’t be surprised if this line freezes solid over the winter and is virtually unusable during those months. The problem is not usually with a major rain on a warm winter day, but rather with a continuous light rain that starts to flow through a pipe within the frost line and gradually coats the inside of the pipe with ice. The exposed end of the pipe through the wall will allow below freezing air to further this problem. A possible solution here is to heat trace the line before you backfill and to turn on the heat when necessary.
I don’t know what sort of soils you have, but if they are at all permeable and if your groundwater table is low, I would consider trying to get the roof runoff away from the building and into an infiltration system of some sort. You can buy some relatively low cost perforated plastic structures or pipes at a home supply warehouse, dig a hole and surround them with crushed stone. A lot of digging, so stock up on the Yuengling.
If your soils are not well drained, here’s a suggestion. Put perforated pipe and crushed stone in the trench as a groundwater collection system. It can be flat if you like. Perforations should face down. If the pipe doesn’t have a sock (a geotextile “condom”), then you will need to wrap the pipe and stone in a geotextile fabric to keep fine soils from clogging the stone and pipe. If the pipe does have a sock, try to mix some sand in the crushed stone to prevent voids that will allow migration of these “fines”. A pipe that only picks up groundwater will have much less concentrated flow, and will probably be much less of an impact in a public way.
Then have your downspouts outfall to concrete splash blocks (as mentioned above by Duckster) to direct the drainage away from your building and toward your infiltration trench. Bear in mind that the best way to keep drainage away from your house is to adjust the surface grades. I like to maintain a six inch drop from the face of the building to a point ten feet away. If necessary (or possible) you can raise the grades at the face of the building, but you should leave a minimum of 8 inches from that grade to any wood parts of your house (check local building codes).
Good luck.
