The only prohibition is owning a vicious dog. Even then if one needs it for protection it is permitted so long as it is properly tied up (Shulchan Aruch Choshen Mishpat 409:3 and see Rema). As for being muktzeh, Rav Moshe says (Iggrot Moshe Orach Chayim 5,22) that if one normally pets and plays with it this is its purpose and it is not muktzeh. However, Sepharadim may hold differently (see Shulchan Aruch Orech Chaim 308:45 with Rema and Mishna Berura seif katan 158 regarding playing with a ball).
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