I'm not going to go through and do all of these time consuming problems, but here are two thoughts with respect to problem 1.
First: In your own attempt at doing the problem, your 15-dimensional irreducible representation is wrong. Actually very wrong. You will never get the right answer with it. Aside from the character of the identity operation, none of your characters are right. (In your defense, Td is tricky because the angles of rotation aren't 90 degrees. You have to use some trigonometry to get the right characters.)
Second: If you had a CORRECT 15D reducible representation for the 3N coordinate transformations, and you reduced this to a sum of the irreducible represendtations for the Td point group, and then you subtracted out the irreps corresponding to translational and rotational motion, you would be left with a total of 9 vibrational modes (3N-6) - some of them combined into degenerate irreps - corresponding to the normal modes of the molecule. This approach to a problem like this will ALWAYS give you the right answer about what all the vibrational modes are - although it can be quite a bit of work to get there.
Third: However the problem is not asking you to do quite this much work. Some of the irreps you would find if you followed the previous process correspond to vibrational normal modes that involve the central atom moving and some of them do not. The question asks you only to identify those involving vibrational stretching motion of the ligands. This is why you don't need to consider the central atom in this problem and it's also why you're not considering a full 3D range of motion of each ligand - only an outward displacement of the ligand from the metal. This simplifies the problem quite a bit and leaves you with a 4-dimensional problem that results in 4 normal modes (3 of them degenerate). These normal modes are a subset of those identified in the more complex problem, and are ONLY the ones that do not involve changing the position of the central metal.
I do have to admit that the intended approach of the problem not obvious to me until I solved the problem the long way. I think it's written in a rather confusing way. Then again, it's been a while since I solved for vibrational modes of polyatomics using group theory, so . . .
Anyway, looking at the other problems: problem 2 also asks you only to consider the four vectoral displacements of the ligands, so this should be solved in a similar way. Problem 3 wants to know what all the symmmetry species are by "displacement of the atoms". Here the ligands are NOT specifically (and exclusively) highlighted so a full treatment is necessary (though again, whoever wrote the problem should have done a better job of making this subtle distinction clear). C2v is an easier point group to work with, so the full treatment isn't quite as tricky as it is in the Td case. Problem four you identify as being treated in a similar way as problem 1. But according to my interpretation of the wording, a full 3D treatment seems to be required. After all it asks for "all symmetry modes". Restriction to displacement of the ligands only is not specified.
So in short, I sympathize. The questions aren't particularly well written and unless you had a lot of experience with symmetry treatments I can see what you were confused. You just need to read carefully what the problems want - if they tell you only to consider displacements of ligands, then you know you don't need to do a 3D treatment of each atom's position. If they want you to solve for every vibrational symmetry mode of a molecule, you know you've got to do the full deal.
Of course, you do have solve for your reducible representations correctly - fail to do this and you won't get it right now matter what approach you take.
EDIT: By the way, the nice thing about these kinds of problems is that it is very easy to check if your reducible representation is correct: simply try to reduce it to a sum or irreducible representations. If you don't get integer values of the irreducible representations, you've done something wrong. Take yours for problem 1 for example (15 2 0 0 2). If you try to reduce this you get 1.79 as the number of A
1 irreps. At this point you know something ain't right.