This isn't identical to, but very similar to a B-tree. It's a generalization on a binary search tree. A binary search tree stores one value at each node, and has two children, with the left subtree guaranteed less than that node, and the right subtree guaranteed greater. With a B-tree, you store k values in sorted order at each node, and allow k+1 children based on in between which two children it lies. This is basically the same thing, except that you have equality instead of inequality (since there are only 26 possible choices at each step). Also, note how in your case you aren't actually storing data at each node, each node just uses the same letters a-z to determine which child to follow down. So you'd likely store children at the leaves. I'm not sure exactly how balance would work out; you'd need to create new nodes whenever you inserted a word that shared the same prefix as another words in terms of nodes you already had.
In your case, you need to give more details so we can answer b). If your list of words is fixed and doesn't change, your best bet might actually be just a simple array. Just sort it once. Then to find all words starting with "foo", you simply do binary search to find "foo" and "fop", or the close possible matches (rounded in the appropriate directions). You then return all the words between them. This will give you logN + numResults in time complexity. It's also likely to have very good constants as you don't have to follow a lot of pointers around. Also has virtually zero space overhead. Obviously, if you are adding words to your dictionary on a regular basis this is not a good solution as adding a word costs O(N), whereas tree solutions will probably cost O(logN) to add a word.