Why do cat (猫) and dog (狗) share the same 犭 radical?
The radical in 猫 actually comes from 豸 zhi4 (beast), which you can still see in the traditional character 貓. 

Whereas the radical in 狗 is in fact 犭 quan3. 

The simplified version of 猫 got the 犭 quan3 radical because of trait reduction and semantic affinity (dog -> beast). 

If it sounds easier to you, you can think of 犭 quan3 in simplified Chinese as generically referring to an animal, as in 狼 (wolf), 狮 (lion), 猴 (monkey), 狐 (fox) and words relating to animals as 猎 (hunt).

As for your other question: 


  1) Why a dog radical has the dog in it with other components?


The dog radical 犭 comes from 犬 which is a pictogram, still used in some words today, especially dog breeds. No radicals other than the image itself. Here you can see an evolution of the picture. (Source: Wenlin, and possibly 说文解字).



If you are talking about the modern character 狗 gou3, it comes from the semantic/radical 犭 quan3 plus the phonetic part 句 ju4. It was probably a colloquial or popular name for "dog" which has made it into the standard language, whereas 犬 keeps a more classical and scientific flavor.