The key to answering your question is to understand that pressure at any depth in a fluid (liquid or gas) causes the same force in all directions. There are several "levels" of understanding this.
The most intuitive way to see it is to imagine sitting on a trash bag full of water. Of course that increases the pressure the water. But even though your weight is pushing downward, water will squirt out anywhere you poke a hole in the bag, including the top, bottom and sides.
But to apply this idea clearly to the bottom of a rock in the water, or the bottom of the table, we need a deeper understanding. Pressure is not force. Pressure causes force (and force causes pressure). Pressure itself is the volume concentration of the energy of the particles (atoms and molecules) in the fluid. It's the energy per unit of volume. In a gas, that energy is mainly kinetic--associated with the mass and speed of the particles. In a liquid, the energy is mainly potential-associated with the particles being pressed a tiny bit closer together against enormous restorative forces, like extremely stiff little springs.
When an object is placed in either a gas or a liquid, the energy of the particles causes force on the surfaces of the object. In a gas the force is mainly the effect of the many, many virtually simultaneous collective collisions of the fluid particles against the surface during any instant of time. In a liquid, the force is the effect of the many, many little "compressed springs" constantly pushing against each other and against the surface, like the famous video footage of a crowd of soccer fans crushing each other against a fence.
In the atmosphere the cumulative weight of the air above compresses the air more and more with increasing depth, so there are more and more particles per unit volume and therefore more energy per unit volume. Near the surface of the Earth the particles are colliding with the top of the table and the bottom of the table at an equal rate. If you put a book on the table, there will be only a tiny volume of air under the book, but the energy per unit volume-the pressure under the book, will be the same.
The weight of the ocean does not compress the water much at all--it's almost incompressible. The tiny bit of compression that does occur does not increase the number of particles per unit volume very much, but it does increase the force between them a lot. So again, the volume of water under the rock may be small, but the energy per unit volume is the same.
It's alright, tension is pretty subtle! Let me answer your questions a bit out of order.
- If you pick any and all points on the rope, would there be two opposing tensions at every one of those points?
- Is tension uniform throughout the rope?
You can think about the rope as a lot of tiny masses connected together by springs; this is a cheap approximation for how tension works on the atomic level, where the springs are stretching chemical bonds and the masses are atoms.
In our simple model of tension, every atom is pulled by a spring on the left and a spring on the right, with forces $T_1$ and $T_2$. Then by Newton's second law,
$$T_1 - T_2 = ma$$
where $m$ is the mass of the atom. Since $m$ is very very tiny compared to the other masses in the problem, we must have
$$T_1 \approx T_2.$$
Applying this to every mass, we conclude that each little spring / chemical bond has approximately the same tension, so we can simply talk about "the tension in the rope". This is a good approximation as long as the total mass of the rope is much smaller than the masses of the blocks.
How might differences in mass between object A and object B (which, sorry if the diagram was misleading in the sizes, can have any mass) play into the tension?
Well, now that we've established that there's a uniform tension $T$, we need to figure out what that tension is. The constraint here is that the rope is taut, which means that it can't be scrunching up or stretching out; that translates to the constraint that the accelerations of the two blocks are equal in magnitude. This equation determines the tension.
How does the relationship between the force of gravity on mass B and the tension in the rope play into this? Isn't the tension caused by that force of gravity? Doesn't that mean that if tensions cancel, the force of gravity's effect is canceled as well?
(I have also been told that the blocks will have the same magnitude of acceleration. Why is this?)
No, the tension isn't equal to the weight of block B, it's whatever is necessary to satisfy the above constraint. For example, suppose that block B was very very heavy, so the acceleration of the whole system will be close to $g$. In this case, the tension is actually quite small compared to the weight of block B, because you only need a little tension to make block A have the same acceleration.
(In fact, as block B gets infinitely heavy, you can show that the tension doesn't go to infinity -- instead, it becomes the weight of block A! It's neat to try to prove this, and see how it works.)
Does the pulley affect tensions? For example, we know that there's a positive tension affecting mass A. Is there still a positive tension in existence on the other side of the pulley, or just the negative tension that's acting on B? Might there be some sort of effect whereby two sets of opposite tensions, one set on each side of the pulley, cancel each other out?
This is a little tricky to word. The tensions of the two tiny springs attached to each atom approximately cancel out, as shown above. But that doesn't mean that the tension is zero -- all of those springs are still stretched.
Since the pulley is frictionless, it doesn't have any effect except that it 'turns around' the tension. You can show this by considering the three forces on each atom (two springs, one normal force) which gives $T_1 \approx T_2$ as before.
Anyway, I've been told that the net force is the force of gravity on mass B (the hanging one). But I don't really understand why, even after extensive discussion with various people.
That's not your fault, the question is just worded badly. There are lots of forces involved in this problem acting on different things: gravity on both blocks, normal on one block, and normal from the pulley on the rope. It's not very clear what "the" net force even means.
Best Answer
During the rowing action, all that is necessary is that an increase in the boat's forward momentum is created by an increase in rearward momentum for something else.
We can easily think of this as a parcel of water. As the oar moves, it's creating a force couple that pushes (accelerates) the rower forward and pushes the water backward.
For the purpose of moving the boat, this is sufficient. But after this action the water interacts with its environment (the lake) and creates forces. The result is that the parcel of water slows down and the rest of the water accelerates backward a teeny bit.
Then that water does the same to its environment. Eventually the forces and momentum of the oar spread out to the lake and the earth. (And then they run in reverse as viscous forces bring the boat to a halt).
So the eventual result is that the rower ends up pushing on the lake/dam/earth, but that isn't required to move the boat. Only the acceleration of the water parcel by the oar is necessary. The rest happens after the oar is gone and doesn't (directly) affect the boat.