May it now be that by definition a free stream has no accelerations? And as derived in the linked page, no acceleration implies no pressure gradients. That with the fact that all the streams considered are connected with the atmosphere at some points, should mean all the stream is at atmospheric pressure.
In the form you've stated it ($A_1 v_1 = A_2 v_2$), the continuity equation is only holds for incompressible fluids. So what you've found is that the type of accelerated flow you're describing cannot happen for an incompressible fluid in a pipe of uniform cross-section. The more general form for the continuity equation is based on conservation of mass (i.e., mass per time entering = mass per time exiting), and states
$$
Q_m = \rho_1 A_1 v_1 = \rho_2 A_2 v_2,
$$
where $Q_m$ is the mass flow rate (i.e., mass per time). This means that if the fluid increases in velocity, it must decrease in density.
An analogy here would be cars on a highway. Suppose you have a highway that leads from the center of a small town out into the country. Suppose further that the drivers are all perfectly safe drivers and obey the two-second rule, i.e., the cars pass a given point in the highway at a rate of one car every two seconds. If the speed limit in the town is low, then the cars will be more closely spaced, since two seconds corresponds to less distance. Thus, the density of cars is higher at this point. When the cars get out of town and the speed limit increases, the cars get further apart in distance (since two seconds now corresponds to a longer distance), and so the density decreases.
Bernoulli's equation, meanwhile, doesn't hold so simple a form for compressible fluids. Rather, you have to define a pressure potential $w(P)$ from the equation of state $\rho(P)$:
$$
\frac{1}{2} v^2 + gy + w(P) = \mathrm{const.},
$$
where
$$
w(P) = \int \frac{\mathrm dP}{\rho(P)}.
$$
For the case of an "incompressible" fluid, $\rho(P) = \rho$ is a constant, $w(P) = P/\rho$, and the familiar form of Bernoulli's Law is recovered. But for a compressible fluid, the equation may look quite different.
Best Answer
Before I start the answer, I'd like to point out that Bernoulli's principle is not applicable directly when you're comparing air flow from two different sources ( or two different flow fields, according to Wikipedia ). It only relates the speed and pressure of air within a single flow field.
Now let's consider the first case where you blow over the sheet of paper. The paper would not rise if it were flat, even though you are blowing air across the top of it at a furious rate. Bernoulli's principle does not apply directly in this case. This is because the air on the two sides of the paper did not start out from the same source. The air on the bottom is air from the room, but the air on the top came from your mouth where you actually increased its speed without decreasing its pressure by forcing it out of your mouth. As a result the air on both sides of the flat paper actually has the same pressure, even though the air on the top is moving faster. The reason that a curved piece of paper does rise is that the air from your mouth speeds up even more as it follows the curve of the paper, which in turn lowers the pressure according to Bernoulli's principle.
If Bernoulli's principle were to hold true in the first case, it would then imply that the paper would droop downward in the second case, when air is blown below the paper. But this is clearly not the case. The upward pressure gradient in this downward-curving flow adds to the atmospheric pressure at the lower surface of the paper. This resulting pressure gradient is the source of lift in the second case. I hope this answers your question.
Reference: Wikipedia
EDIT - A formal derivation of the mathematical relation between curved streamlines and pressure gradients can be found here: https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/fluid-mechanics/f20_fall.pdf
EDIT 2 - Lift is also generated by the Coanda effect. Here a couple of links:
How is lift generated due to Coanda effect?
http://en.wikipedia.org/wiki/Coand%C4%83_effect