Dear Jenkins, the theories you want to construct are "noncritical string theories" and they're less interesting and less consistent than the "critical string theories".
First, the Nambu-Goto action - the proper area of the world sheet - is nonlinear. It includes square roots etc. It's much better to introduce an auxiliary metric tensor on the world sheet and the action for the coordinates $X$ becomes nice and bilinear - a free theory.
However, we don't want new degrees of freedom to be added. The 2D metric tensor has three independent components. Two of them may be set to a standard form by the 2 degrees of freedom in the 2D coordinate reparameterization symmetry; and the third by the Weyl symmetry if it exists.
If it doesn't exist, it's too bad. The auxiliary world sheet metric may only be brought to the form of $e^\phi \eta_{ab}$. That means that $\phi$, determining the overall scaling, becomes another function of the world sheet coordinates $(\sigma,\tau)$, very analogously to the spacetime coordinates $X(\sigma,\tau)$. In fact, it is really valid to say that the parameter determining the overall scaling of the metric is another spacetime coordinate.
If this coordinate were totally identical to the other coordinates, then there would also be a translation symmetry in the $\phi$ direction - but that's equivalent to the Weyl symmetry (multiplicative scaling of $e^\phi$ is the same thing as additive shifts to $\phi$). Because by assumption, the Weyl symmetry doesn't hold in your theory, the new spacetime coordinate $\phi$ can't have quite the same properties as the other spacetime coordinates.
However, in normal circumstances, you obtain the violations of the Weyl invariance as a disease. In particular, if you try to study string theory in a non-critical dimension, i.e. $D\neq 26$ or $D\neq 10$, you will find out that the field $\phi$ doesn't decouple and the path integral, when calculated including the one-loop accuracy, still depends on $\phi$. So the Weyl symmetry, equivalent to an additive shift of $\phi$ by a function of the world sheet, is not a symmetry.
As I said, this can be interpreted as $\phi$'s becoming a new spacetime coordinate. But if you try to calculate the effective action in the new spacetime that has an additional dimension $\phi$, you will find out that the laws of physics are not invariant under translations in $\phi$ - that's nothing else than the failure of the theory to be Weyl-invariant.
In particular, you will find out that the dilaton linearly depends on $\phi$: search for papers about "linear dilaton". The squared gradient of the dilaton is related to the surplus or excess (if it is time-like or space-like) of the spacetime coordinates, relatively to the critical dimension.
If the spacetime has two dimensions, one may choose the dilaton to depend on the (only) spacelike coordinate $\phi=X^1$ in such a way that the theory including $\phi$ is Weyl-invariant again. In this case, it's useful to consider not only the right linear dilaton - solving the equations of motion - but also a non-trivial background for the tachyon. One ends up with the so-called "Liouville theory" - a "linear dilaton" theory with some extra tachyonic profile in a non-critical stringy $D=2$ spacetime - which is slightly more consistent than other noncritical string theories. The Liouville theory may also be described by a quantum mechanical model with a large matrix - the old matrix theory.
It's somewhat unclear how you "understood" and are "happy" about the definition of the conformal transformations because your questions, while referring to page 2 and other things, are nothing else than misunderstandings about the definition of a conformal transformation which is explained on page 1, not 2.
John wrote his equation 1 which states that conformal transformations "are" diffeomorphisms that only change the metric by a local scalar coefficient - by a Weyl rescaling. However, the invariance of a theory under these diffeomorphisms is a trivial property. If one is allowed to change the terms coupled to the "metric", a diffeomorphism-transformed theory has a different action in general, and it is always possible to rewrite the original action in the conformally transformed coordinates.
But what's physically nontrivial is the condition that the action, in its original form, is actually invariant under the operations - that's what we mean by the theory's being conformally invariant. If a theory is conformally invariant, we don't allow any "change of the coefficients" in the integral of the Lagrangian density. This condition of conformal invariance, as he shows, is equivalent to the invariance of the theory under the "Weyl rescaling" only: we just completely eliminate the diffeomorphisms from the picture.
So:
Again, the invariance under the combined "diffeomorphism" and "Weyl rescaling" (the latter changes the form of the action) is a tautology. Obviously, by conformal invariance, we don't mean a tautology, so by conformal invariance, we mean the invariance of the action under the diffeomorphism separately, without changing the form of the metric in the action. Because the invariance under the "combo" is tautological, conformal invariance is equivalent to the invariance under the "Weyl rescaling part" of the transformation only.
No, on page 2, the transformations are exactly what the equations say: point-dependent transformations of the metric tensor itself i.e. a Weyl rescaling. There is no diffeomorphism at this stage. The invariance under those Weyl transformations is equivalent to the invariance under some diffeomorphisms (with the modification of the metric erased), as explained in the previous point.
The formalism may depend on classical physics but your comment that it is "classically only of course" is incorrect, too. All those facts about conformal transformations are completely valid quantum mechanically as well - and indeed, this background is primarily mean to understand some quantum theories.
Best Answer
The two-dimensional Polyakov action for a string with worldsheet $\Sigma$ and worldsheet metric $h_{ab}$
$$ \frac{T}{2}\int_\Sigma \sqrt{-h}h^{ab}g_{\mu\nu}\partial_aX^\mu\partial_bX^\nu$$
has full conformal symmetry under the Virasoro algebra and under Weyl transformations1 , which can be seen as gauge degrees of freedom. It follows that we can always treat the worldsheet metric as being flat up to a Weyl factor (or even without it). Consequently,
$$h_{ab} = \mathrm{e}^{2\phi} \eta_{ab}$$
is known as the conformal gauge. It should be noted that Weyl symmetry is special to two worldsheet dimensions, since the action is not invariant under Weyl transformations for other dimensions due to $\sqrt{-h}\mapsto(\mathrm{e}^{2\phi})^{D/2}\sqrt{-h}$ in $D$ dimensions under a (local or global) Weyl transformation $h_{ab} \mapsto \mathrm{e}^{2\phi}h_{ab}$.
1For the difference between the two, see this question and answer.