Forgive me if I'm telling you things you already know, but I'll mention it for others, if not for you. Erosion is caused by two things - volume and speed.Anything you can do to reduce either one reduces erosion. The lawn is good because it has a lot of blades that the water has to go around which slows it down and also allows the soil to absorb some (both reducing speed and volume). Also, slope translates to speed. Longer slopes translate into increasing speed. Anything you can do to disperse water rather than concentrating it is reducing volume. Sometimes we combine drainage so that we deal with it in fewer places. But sometimes it is not a good idea because you create a big problem instead of several smaller and easier to deal with problems. Since you mentioned that the path is a convenient way to divert water, it makes me think that it is intercepting water that might be moving across the lawn and concentrating it (making a bigger volume) more or less in a channel that is the path. If it goes downhill on a steady slope, water will build up speed and erode. Interupting the slope with a step every once in a while will flatten the grades between the steps, thus slowing down the water. If the path has a cross slope, the step can divert the water off of the path at that point removing it from joining up with the water that the path is accepting between that and the next step. You may have seen this technique on hiking trails just using logs. Even if the water has to pass over the step, it will be reaching the edge of the step with very little speed. although dropping over the edged of a step will add speed to the water, it has ery little time to accelerate and lands on another slow slope. It never builds momentum. Another good technique is to privide better drainage closer to the source. It seems that you hinted that the driveway may be the biggest source of runoff. I will assume that the water is not channeled in a paved swale that concentrates it, but rather drifts off onto the lawn all along its path. In that case, a narrow deep trench filled with larger stone can hold some volume and provide side and bottom area for the soil to absorb some of it. I would not continue such a trench where the driveway slopes a lot as water would rush along the trench. A retention swale that holds volume and traps it untill the soil absorbs it is an option, but most people don't like to have a puddle or wet area. Some have the space for it and don't mind. Another thing is a detention swale that captures the volume and holds it temporarily by having it drain out in a limited manner such as making the swale drain out of a small pipe that can only handle so much water. It detains the water only to let it go on its way in a controlled release. Compaction of your stone in the path is not really much of a problem for water erosion. Smaller surface stone will wash with less force than larger surface stone. Stone in swales is often refered to as "energy dissipation" material where it takes the force of fast concentrated water coming off of a pavement or from a pipe. It takes the impact and slows the water down in the initial portion of the swale. The slope of the swale will then take over in managing the speed of the water and the stone is no longer necessary. Hopefully, this will give you some insight to help you analyze what is happening on your site and some different ideas on what might be options for your specific situations.

You mentioned that the bottom is sloped and that's a good start. I'm assuming that every point in the bottom of the planter slopes to an outlet - the more outlets the better. I'm also assuming that the slope is, at the very least, a 1.5% grade. If not, you'll have problems. Think about where the drains are and the shape of the planter. For example, if you have a single drain at the lowest spot of an 'L' shaped planter, water can get hung up at the bend. You may have to drill out more drains depending on the size and shape of the planter. The fabric can indeed keep your soil in your planter, but over time it will plug with soil that you're keeping in and drainage will be a thing of the past. Once it's in, there's no way to go in and clean out the fabric. The gravel may help with drainage at first but over time, soil will sift down and maybe next year, the benefit of the gravel will be gone. The key to a long-lived planter is to keep water moving over the long haul. If the bottom of the planter gets mucky water won't move and you'll end up with anaerobic soil conditions and/or an environment that favors the pathogens responsible for root diseases. Silt and clay will end up settling to the bottom and doing just that if you use a good old fashioned "import soil" which will typically fall in the sandy loam category. Your best bet is to use a high quality 'medium' sand with less than 6% silt and clay. Organic amendment will help improve nutrient and water holding capacities. However, organic matter below ten inches in the planter can also muck things up and lead to an undesirable amount of settling as the O.M. decomposes(I would only put it down to 6 or 8 inches but other go a little deeper) so if your planter is deeper than 10 inches, fill the bottom with sand and then include organic matter in the top and do not exceed 80% mineral soil to 20% O.M. by volume. Keeping the organic matter% relatively low in the sand will keep the water moving and also avoid settling. The O.M. will decompose over time. If that O.M. is placed deeply and/or at too high a volume, the settling can result in exposed roots and planter flashing. I don't know where you are or what's available to you, but redwood bark is a nice long-lasting source of O.M. for these types of projects.

perforated pipe in a trench with a slight pitch toward the back yard, covered with landscape cloth that can be penetrated by water, then covered with pea gravel, road base, or the like. right along the fence line. the rest of the concrete walk just break up and replace with new, but slope just slightly from the house (up side) out to the place where the pipe lies so water will run away from the house rather than pool and run down the sides of the foundation. you may need to do something like a French drain at the back where the water from the pipe will flow. if you don't like the look of regular concrete sidewalks you can get them molded or stamped to look like almost any surface, get them stained, imbed things in the concrete, etc. best wishes

I will second Yardvaark's comment on the pea gravel. It moves underfoot rather like ball-bearings, gets inside sandals, and can be kicked by normal walking into gardens, lawns, etc. A surface that includes fines and so packs into a solid-feeling surface, whether something like what is referred to as crusher-run around here or shredded bark will be far more comfortable and functional. Metal edging (though probably best this can difficult to come by for homeowners IME), Trex type composite edging if well supported from both sides to prevent warping, or well-buried plastic edging all are possibilities.