Why is a hot air balloon  "stiff"? 1) Why is a hot air balloon stiff? 
2) Is the pressure inside the balloon higher than the pressure outside (atmospheric pressure)? 
3) If the pressure inside is higher than the outside, how is it explained by ideal gas law?
 A: To understand the physics of inflated balloons you have to understand curved membranes under tension.
If there's a membrane under tension $T$ (for simplicity assumed to be isotropic), with fluids on both sides, it will be flat (planar) unless there is a pressure difference between the two fluids. If you notice that it's curved with radius of curvature $R$ (again assumed to be isotropic, like a sphere), then there must be a pressure difference given by
$$\Delta p = 2 \frac{T}{R}$$
Of course, the membrane bulges away from the side where the pressure is higher, out into the side where the pressure is lower.
Now that we have this basic concept, the answers to your questions all come from it because they're all related.
The hot air balloon is stiff because it is under tension. If a membrane is not under tension it could have folds or wrinkles because there is no energy penalty for having extra area. But since the material of the balloon is under tension it's going to be as smooth as possible. It would be flat if not for the pressure difference, which forces it to have a smooth curve.
The pressure inside the balloon is indeed higher than the pressure outside, and the difference is given by the formula above.
This pressure difference is not really "explained" by the ideal gas law, but it is consistent with it. If you start with a deflated balloon on the ground and gradually fill it up with hot air, you're increasing $N$ (because there are more air molecules inside the balloon than when you started) and $V$, while $p$ and $T$ remain roughly constant (the $T$ inside being higher than the $T$ outside). When $V$ finally gets to the volume that will make the balloon stiff, the pressure begins to go up slightly above the ambient pressure, with the extra pressure being provided by the tension of the balloon. However, the density of the gas inside the balloon is still less than that of the air around it, because although $p$ is greater (tending to increase the density), $T$ is also greater (tending to decrease the density), and the temperature effect dominates.
A: I'm going to try to accomplish something the other comments don't quite get.
The challenge here is that we're really talking about a fluids problem, really a hydrostatic problem.  You can't just write the ideal gas equation and be done with it, although no one has really written that equation yet, so I'll write it here.  But I'm not just going to write it.  I'm going to write it in the molar / volumetric form that I personally believe does the most to answer the question.
$$ \frac{P}{\rho} = R_{\rm specific}T $$
$\rho$ represents density and $R_{specific}$ is constant.  I will stress the point that the total number of molecules in the balloon doesn't matter.
We have established that, yes, the pressure is higher on the inside of the balloon, as evidenced by the observed rigidness.  There is nothing in the above equation that gives us an idea as to why.  We need to introduce something else.  That something else is the fact that the balloon exists in a gravitational field.
(I would add in response to the prior answer that it's not about the "gradient" of $P$ or $T$, but hopefully the rest of my answer will make it clear why)
A non-constant temperature within a gravitational field leads to pressure differences.  Specifically, let's talk about going from point $A$ to point $B$.  Again, keeping with the hydrostatic assumption, we can fairly easily establish the following for the difference in pressure between the two.
$$P_{A} - P_{B} = \int_A^B \rho(\vec{r}) g d\vec{r}$$
This is my version for the common $\rho g h$ that gives the change in pressure over a certain column of a fluid.  I will reduce it back to these terms for the specific case of $A$ being a point on the inside surface of the balloon and $B$ being right next to it on the outside.  Another assumption that I make is that the temperature is perfectly constant inside the balloon, $T_b$ and perfectly constant outside the balloon, $T_{air}$.
$$P_{A} - P_{B} = - (\rho g h)_b + (\rho gh)_{air} = g h \left( - \rho_b + \rho_{air} \right) $$
$$ \rho = \frac{P}{R_{specific} T}$$
$$P_{A} - P_{B} = \frac{g h P}{R_{specific}} \left( - \frac{1}{T_b} + \frac{1}{T_{air}} \right)$$
Were you confused by the signs?  I certainly was.  You start at $A$ in the balloon and go downward, so the term for the path inside the balloon should be given a negative.  Next, why did I take $P$ constant?  Doesn't that violate the entire point of the problem?  No.  Pressure has little impact on the density in this case, but this is a relative statement, and it's relative to the temperature contribution to the density difference.  The temperature within the balloon could be $10^{\circ} F$ higher than outside, which would make the temperature difference on the order of $5$% relative to absolute zero.  The same is not true for pressure.  The pressure difference due to the entire hydrostatic head for the entire balloon height is roughly $0.01 psi$, or $0.1$% of absolute.
A: I have an explanation for this. I would like to verify if that makes sense. 
Basically my hypothesis is that whenever there is a temperature gradient, there will be a pressure gradient. In our case there is a temperature gradient between outside the balloon and inside the balloon. This leads to a pressure gradient. This pressure gradient (read pressure difference) causes the balloon to be stiff.
Please refer to http://rejeev.blogspot.com/2011/03/temperature-and-pressure-gradient-in.html for details.
I would like to seek feedback from the community on my hypothesis.
