Hungry is indeed hungrig.
Just "someone around" (don't you mean "someone is around"?) is indeed "jemand ist in der Nähe", although it is not too common. If however, you mean it as a question, something like Is someone around?, then this would be Ist jemand da? (usual one) or Ist jemand in der Nähe?.
Cooking is indeed kochen, so for example He is cooking translates to Er kocht.
Go get some would then translate to Gehe los und hole etwas, for example when you want to say to your friend Go get some bananas, that would be Gehe los und hole Bananen.
I hope this helps you.
Your translations were not bad, you are on a good way of learning German!