We notice sudden changes in anything. We don't notice gradual change, whether in time or spread over space. If the car is moving uniformly along straight flat road, its acceleration a=0. Its velocity v is constant.
When the brake pedal is pushed, friction causes the car to decelarate. a = some negative number. You can't avoid that. You want to stop? Your velocity must decrease. (Duh.)
What the passengers feel is the change from a=0 to a=-5.6948 m/s^2 (making up a number.) You want that change to be gradual. Before braking, when a=0, and assuming you're sufficiently skilled and anticipate needing to stop, you light apply brakes. Suppose you make a linearly decrease from 0 to -5.6948, and once there, keep it there. This linear decrease takes place over, let us say, 5 seconds. Likewise, a linear let-up to a=0 timed perfectly as the car comes to a stop.
Remember acceleration is the time derivative of velocity. Velocity, therefore, is the indefinite integral of acceleration, which is a piecewise linear function of time. Plotting velocity vs time, you'd see pieces of parabolas joined together.
Now, I wonder if passengers would feel the change in acceleration during that time it's changing? It's changing gradually, but it is going from non-changing (zero) to changing linearly, to finally not changing (steady strong braking), with no sudden jumps. Maybe the game is to soften the rate of change of acceleration, to not suddenly go from flat to linear. Look at the time derivative of acceleration - it's all step functions in our example!
So make the time derivative of acceleration be piecewise linear, no jumps. Then the acceleration is smooth, no sudden changes in slope where parabolas join. Velocity is very smooth, described by 3rd degree polynomials. A good driver/pilot approaches this kind of smoothness by intuitive feel and experience.
1) You surely feel the pressure when you accelerate. Whether you attribute it to fictitious forces or other forces depends on your choice of the "reference frame" (vantage point). From the viewpoint of your body's reference frame, which is not an inertial frame, there exist fictitious forces (inertia and/or centrifugal and/or Coriolis' force) that are pushing your body towards the seat. In an inertial reference frame, such as the vantage point of people who stand on the sidewalk and watch you, the pressure is exerted by the seat because it's accelerating i.e. pushing (you are pushing on the seat as well, by the third Newton's law) and there are no additional fictitious forces. Both of these descriptions are OK but the description from the inertial systems (e.g. the sidewalk system) is described by simpler, more universal equations. Without a loss of generality, we may describe all of physics from these frames and these frames never force us (and never allow us) to add any fictitious forces. The frame of your (accelerating) body may be considered "unnatural" and therefore all the forces that appear in that frame are artifacts of the frame's being unnatural, and therefore they are called "fictitious". They may be avoided.
2) Centrifugal forces are the textbook examples of fictitious forces; they have to be added if you describe the reality from the viewpoint of rotating systems. They are avoided if you use non-rotating frames. However, the tides have nothing to do with centrifugal forces. The tidal forces appear because the the side of the Earth that is further from the Moon is less strongly attracted to the Moon than the side that is closer to the Moon. In other words, the tidal forces totally depend on the non-uniformity of the gravitational field around the Moon – the force decreases with the distance. You could create the same attractive force as the Moon exerts by using a heavier body that is further than the Moon. The attractive i.e. "centripetal" force would be the same but the tidal forces would be weaker!
3) In an inertial system – connected with the Earth's surface, for example – the force acting on you is $mg$ downwards from the Earth's attraction plus $ma$ from the extra upwards accelerating elevator. The part $ma$ has a clear new source, object that causes it, namely the elevator. However, in a freely falling frame, for example, the gravitational downward $mg$ force cancels against the fictitious inertial force $mg$ upwards. However, the material of the elevator is now accelerating by the acceleration $g+a$ upwards so the total force is $m(g+a)$ again.
As you can see, whether there are fictitious forces depends on the reference frame. What I feel is your trouble is that you're not used to describe processes from the viewpoint of inertial reference frames. Take a spinning carousel. There is a centripetal force acting on the children and this force, $F=mr\omega^2$, is the reason why the children aren't moving along straight paths with the uniform velocity (as Newton's first law would suggest). Instead, they're deviating from the uniform straight motion and move along circles. The centripetal, inwards directed force $mr\omega^2$ from the pressure from the seats is the reason. (For planets, the centripetal force is the gravitational one.) There are no fictitious forces, in particular no centrifugal force, in the description using the inertial system (from the viewpoint of the sidewalk). However, from your rotating viewpoint, there is a centrifugal force $mr\omega^2$ acting outwards that's always there because the frame is rotating. This force is cancelled against the pressure from the seat, a centripetal force $mr\omega^2$, and the result is zero which implies that in the rotating frame, the coordinates stay constant in this case, especially the distance $r$ from the axis of the spinning carousel. Both frames are possible: one of them forces you to add fictitious forces, the other one (inertial frame) doesn't contain any such forces.
Best Answer
I have to record this comment by user Velut Luna for its pithy logic:
which is certainly true, but there is another sense wherein jerk can directly affect our bodies in some cases. Those cases are when one's body is accelerated through the reaction force between the body and a "thrusting" object, such as the seat of a car undergoing acceleration. Our bodies are deformable, and not all parts of them accelerate in the same way: the seat thrusts the parts of the body in contact with it, and these deform. It takes some time for that force to be transmitted through the tissue in direct contact with the seat to the tissue furthest away from the seat. Therefore, accelerations with different jerk as a function of time will give rise to different strains / stresses in the body as a function of position and time.
The same is not true if the body is accelerated by a body force, such as, for example, if it were uniformly charged and accelerated by an electric field. All volumes would undergo exactly the same acceleration so no internal strain would result, whatever the acceleration or jerk or any higher derivative may be. See my answer and the companion answer it references here.