の is a verb normalizer. Unlike in English, Japanese can't use a verb (する here) as a subject or object to another verb (やめでください here). That's why の is used. の turns the first part of the sentence (walking every day) into a noun, so it can be used as a direct object.

こと「事」 is another verb normalizer (which means 'thing'), so using こと here would make the sentence literally: "Please do not stop the (walking every day) thing".

For extra information, any noun can be used here. So for example: しりません人 (a don't know person) would mean: "a person (I) don't know". This is different to 人をしりません, which means "(I) don't know the person."

I guess because "aruku" would imply only walking any amount (which everybody does unless they're physically disabled or injured) whereas "sanpo suru" implies a whole Walk, outdoors, which the English sentence probably entails (though it's not unambiguous).

I guess it's like "renshyu suru" Renshyu is practice when on its own where as when it's with suru it becomes the verb "to practise". Sanpo suru becomes "to take a walk".

やめる is transitive and takes an object naming the thing that is being quit (in this case, the walks). If you used the subject particle が instead of を, it would mean the walks themselves were quitting something.