Can I repair chips and holes in a wall with spackle and paint?
Yes spackle can fill those places. Do remove the loose piece at the bottom before you fill. The rough texture on the walls will make it difficult to get you patch smooth, and I don't think sanding the spackle will be the way to go when the spot is filled. There will always be overlap of the spackle when filling a low spot, but try to keep it to a minimum. After the low spot if filled sufficiently, and it may take  few coats, use a damp or wet sponge if needed to wipe the excess off of the existing texture to minimize the texture change. The roughness over the new spackle may be left alone, since sanding it flat may be more noticeable than the roughness left naturally by the application.
For the hole in the ceiling, that same thing can be done, use a small putty knife or even your finger to fill the hole, and wipe the excess with a damp sponge.