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First, the kick doesn't end until someone possesses the ball (i.e. if R muffs it, it is still a kick).
For situation (a), K can catch or recover ANY scrimmage kick in or behind the neutral zone and advance. This applies even if the kick went beyond the NZ and then came back behind it. It also applies if R muffed the kick beyond the NZ and the ball bounced back behind the NZ (remember, the muff doesn't end the kick). Rule 6-2-3.
When K recovers a kick beyond the NZ the play ALWAYS becomes dead as soon as K recovers, as per rule 6-2-4. As for the specific case book play, K gets a new series in both (a) and (b) because if R is the first to touch the kick beyond the NZ, K will keep posession if they recover (no matter where K's recovery occurs). This is from rule 5-1-3f.
In the situation of a fumble, the kick has ended since a team has gained possession (there must be possession to have a fumble). This fumble is just like any other fumble, and may be recovered and advanced by anyone. Rule 7-4-2.
Hope this helps.
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