The growth along the sides and in the cracks has similar causes. The side-growth typically results from grass growing over the sidewalk. This grass traps wind-blown soil and organic particles and, as it ages and dies, the blades contribute to the organic material under the living growth. Over time, you get soil, which then allows the grass to grow even better. The growth in the cracks can be infiltration from the edges along with weed seeds that sprout in the wind-blown soil particles and organic material that naturally builds up in crevasses.
Some of those sidewalks look to have been unused for years, especially those pictured in photos 3-5. If they aren't ever used by you, you could simply add some topsoil to them and encourage the grass to continue growing over them. In times of drought that grass would go dormant first, but otherwise it would look like the rest of your lawn. I have seen this in the older portions of my own small city (example: one guy wanted to expand a flower bed at the front of his house and when he began manually trenching for the edging ran across a sidewalk that was three inches below grade).
Otherwise, your plan to remove the grass and soil along the edges will work; doing it manually will be a good workout, or you can buy an edger. There really is no way to know how many power edger blades you'll go through without knowing how many linear feet you have to edge. Some folks may opt for a string trimmer, but that may require many string replacements. If you go the manual route, buy a half-moon edger and sharpen the blade as needed. Cheaper than a power edger, but more work, of course.
As for the growth in the cracks, you may have to resort to a hand-trowel and/or a Korean tool called a Ho-Mi. As you can see via the link, this tool has a narrow pointed tip that's good for getting things out of cracks. I don't think you'll be able to get that growth and soil out without resorting to working on your hands and knees for more than a few hours.