Fictitious forces confusion
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.