Is force really considered a push or a pull? If an object is pushed/pulled at a constant velocity and it doesn't accelerate, is there a net force still being applied to the object? A force is a push/pull so there must be a force, despite there being no acceleration, right?
 A: Per Newton's 1st law an object at rest or moving at constant speed in a straight line remains at rest or moving at constant speed in a straight line unless acted upon by a net force.
Per Newton's 2nd law, the acceleration $a$ of an object is
$$a=\frac{F}{m}$$
where $F$ is the net force acting on the object.
It follows then from both laws that the net force on an object moving at constant speed in a straight line (zero acceleration) must be zero.
The above does not necessarily mean no force (push or pull) is acting on the object. It's just that there is an equal and opposite force (pull or push) somewhere also acting on the object.
Hope this helps.
A: If the object has constant velocity, it isn't accelerating. Because $a$ = $\frac{F}{m}$, $\frac{F}{m}$ must be equal to zero. Multiplying both sides by $m$, we get that $F = 0$, so there is no net force. Each force causes a "phantom acceleration," and the real acceleration is the vector sum of these phantom accelerations. Similarly, the net force is the vector sum of forces, and considering the forces independently, they each provide a phantom acceleration in their direction, but there is no net force if and only if there is no "real" acceleration. It is possible to push or pull something at constant velocity if there are other forces acting on it, but if your force is the only force (which in reality could never happen due to gravity from everything in the universe), then the phantom acceleration caused by the force you apply is the only phantom acceleration, so it is the real acceleration, and since the force you apply is nonzero, the object changes velocity. It is possible to push or pull something with constant velocity, like pulling a tug-of-war rope or pushing a box across the ground, but each force causes a phantom acceleration, which is evident when you stop pushing the box and it decelerates.
A: Newton's 2nd law always holds true:
$$F_\text{net}=ma.$$
So, if you have constant velocity, meaning zero acceleration, then no net force is present, $F_\text{net}=0$.
But that doesn't mean that no forces are present. The net force is just the sum of all forces. Maybe you are pushing on the cart while a friction force is counteracting your push, so that their net force sums out to zero.
Whether you want to call a force push/pull is more of a subjective matter. Sure, you pushing the cart is obviously a pushing force, but what is a friction force? A pulling force? Ah well, this is just semantics and doesn't matter. As long as you are able to correctly identify the directions of forces that are involved, then this wording can stay ambiguous.
A: Push / pull is the first intuitive notion of force to me. Later on, by using the elasticity property of materials, this notion can be quantified, and force is measured according to some displacement.
At a third stage, we note that for some controlled situations, a measured force is related to mass $\times$ acceleration. We now redefine force as something acting on a body, when it is accelerated. It has the advantage for example to help to calibrate springs and other devices that measures force by some displacement.
The side effect of that redefinition is that any object subject to a measured force, that doesn't accelerated as expected by the Newton's second law, must have by definition some other force, even if it can not be directly measured .
Examples:
An elevator moves with a constant velocity. The force on its cables can be measured by load cells. The force of gravity is postulated to explain the absence of acceleration. So, the net force is zero.
A heavy object is pushed horizontally, but doesn't move. The force acting on it can be also measured by a load cell. But the absence of acceleration obliges us to postulate a friction force, even if it can not be directly (that say by other way than merely by difference) measured. Again, the net force is zero.
A: If you're pushing an object and it's not accelerating, you may confidently deduce that there is another force, acting on the object, of equal magnitude and opposite direction opposing your push.
This isn't Newtons 3rd law. The 3rd  essentially says that the object itself pushes back as it accelerates. But if the object isn't accelerating, external forces on the object are in balance.
A: If you push or pull an object to exert a force on it, and there are genuinely no other forces acting on it, then it is impossible for you to push/pull such that it moves at constant velocity. If there is a net force, there must be some acceleration; if there is no acceleration then there must be no net force.
You won't be able to test this because you don't have access to a frictionless void where there is no gravity (nor even to an environment that could approximate this, such as free fall in outer space). Your intuition about force and acceleration might be misleading you, because you have never in your life pushed or pulled on an object that wasn't also being affected by other forces.
In particular, the thing that often comes to mind is something like pushing a box across the floor. We are used to the obvious fact that you have to keep pushing to keep it moving, when physics tells us that if the box were pushed (with no other forces) it should accelerate, and then if we stop pushing it should keep moving at constant velocity. Obviously that's not what happens!
The main thing missing from this picture is that friction is a force that opposes motion. While the box is moving there is a friction force "pushing" it backwards, which will quickly decelerate it. So you have to keep pushing it with a force that balances out friction in order to have a net force on the box of zero, which will allow it to move at a constant velocity. Physics says that an object under absolutely no force at all behaves exactly the same as an object under lots of forces that all balance out to zero. So you pushing on it to oppose friction makes it behave like it being affected by no force; it does not behave like it is being affected by only your force.
It might help intuition if you consider pushing something on wheels, instead of a box. There are still forces opposing the motion, but they take much longer to decelerate it to zero. So you can see that a single push is enough to keep the wheeled object moving for quite a while after you stop pushing it. Hopefully that is enough to let you imagine the idealised case where there wasn't any other force like friction opposing the motion, and it would just coast forever. In that imaginary scenario if you were able to catch up with it and push it some more while it was travelling it would just accelerate (until it moved faster than you and you couldn't keep pushing it); there would be no way to "push it at a constant velocity".
You can only "push things at constant velocity" when something else is pushing back. Commonly the thing "pushing back" is friction, but if you're pushing something uphill gravity is also pulling down and you have to overcome that; in that case you can have a case where you need to keep pushing something just to hold it still! (In fact the "keep pushing to hold something still against gravity" scenario happens much more mundanely every time you hold something off the ground)
A: Newton's second law of motion is that force equals mass times acceleration (F=ma). This means that acceleration is a necessary part of force. No acceleration, no force. Velocity doesn't matter.
Newton's third law neatly explains your near epiphany here: for every action in nature there is an equal and opposite reaction. You lean on a wall, the wall "leans" back, and neither you nor the wall move. All the force you apply leaning can be traced through the timbers and into the earth, which you are standing on, so you're essentially leaning on yourself. This is the equal and opposite reaction. Without the timbers supporting the wall the opposite force works in your favor. The opposite force is still in your feet, but is met with Earth's mass, which gives way nearly nothing because of its size. You lean on the wall, it accelerates as long as you lean (a=F/m), then it continues moving before gravity takes it, another force, taking your applied force to zero.
In open space, when you push your friend who's equal to your mass, you and the friend accelerate equally away from each other during your push, but then no longer accelerate once you part, however you both now have equal velocity, but in opposite directions.
Back on Earth, "pulling at a constant velocity" is a perfectly sensible statement that intuitively has you thinking about a constant force to accomplish this. A truck pulls a trailer, keeping velocity, and is definitely applying force. But force is mass times acceleration, and the truck is not accelerating! Technically, it is. It is accelerating just enough so that the force equals the force of gravity and friction which are pulling the trailer in the opposite direction. We notice this opposite force when the truck stops pulling and the trailer decelerates. It's when the force applied overcomes friction and gravity is when the truck actually gains velocity, but regardless of practical gains or losses in velocity, acceleration is always present when force is applied.
