A garden wall gets weather on both faces and on the top, which is why it deteriorates faster than the house it belongs to. Most of them fail at the coping, because once water gets into the top of a wall it works down through it and freezing does the rest.

A leaning wall is often a foundation problem or a tree root rather than a masonry one, and rebuilding it without dealing with that just resets the clock.
Rebuilding from a course that is still solid rather than propping up something that is not.
Reclaimed where it suits, so a rebuilt section does not stand out against the rest.
The top is what protects everything below it. Properly bedded coping with a drip or an overhang so water is thrown clear of the face.
Lime on older walls, a cement based mix on modern ones, finished to match what is there.
Send a photograph of the problem and you will get a straight answer on what it needs, at no cost.
Usually not for a like for like repair, but height limits and listed or conservation status can change that. If the wall is shared, the Party Wall Act may apply and that is worth checking before work starts.
Usually, with reclaimed brick. Matching is about size, colour and how weathered they are, and a new brick next to a hundred year old one always shows.
Because it is the only horizontal surface. Water sits on it, gets in, and freezing pushes the masonry apart. Good coping is the best thing you can spend money on for a garden wall.
Yes, in brick or stone, with proper foundations and coping.
Not always. Sometimes a section can be taken down and rebuilt. If it is leaning because of a foundation or a tree, that has to be dealt with either way.