Special relativity and general relativity have different views about inertial frames, but in some ways the general relativity take on them is (perhaps surprisingly) easier to explain. So I'll start with GR then extend the description to SR.
In general relativity there are usually no global inertial frames i.e. it is impossible to construct a frame that looks inertial everywhere. However it is always possible to construct a locally inertial frame. This is a frame that looks inertial as long as you consider only the spacetime in your immediate vicinity. To check if your frame is locally inertial you surround yourself with a sphere of test particles then watch to see what happens to them.
If the sphere of test particles remains unchanged in relation to you then your frame is locally inertial.
However if the sphere of particles moves with respect to you then your frame is non-inertial because it's accelerating. Finally if the sphere of particles changes shape then the frame is non-inertial because there are tidal forces acting.
Now see how this definition applies to your two specific questions:
if you and the test particles are falling down an elevator shaft then you and the particles both accelerate with the same acceleration of $g$. That means your frame is locally inertial. Exactly this reasoning applies to an astronaut aboard the International Space Station. Suppose you're in a part of the the ISS where there are no windows. If you throw a ball it's going to travel in a straight line at constant speed, even though the ISS is moving in a circular orbit around the Earth. If you don't look out of the windows you couldn't tell you weren't floating freely in space far from any masses.
If you're in a rocket hovering at a fixed distance above the Earth then when you let go of the test particles they'll fall to the floor. This is an accelerated frame not a non-inertial frame. If you throw a ball it won't travel in a straight line at constant speed.
So actually the answers to your questions are the other way around to what you thought.
Incidentally, if you look more closely at the falling elevator frame in your question (1) you'll realise it isn't inertial either. The acceleration due to gravity changes with distance from the centre of the Earth, so the test masses nearer the centre will accelerate slightly faster while the ones farther from the centre of the Earth will accelerate slightly slower. The result is that your sphere of test particles changes shape and gets stretched out into an oval. This is an example of tidal forces.
However if the size of your test sphere is small the tidal forces will be small. If we make the radius of the sphere really tiny the tidal forces will be undetectably small. This is what we mean by locally inertial - the frame looks inertial if we consider a small enough region of it.
Now let's consider special relativity. In this case there is no gravity so if you were in the elevator shaft you'd just float there. In this case your sphere of test particles would not move with respect to you no matter how big you made the sphere. So this frame is inertial, but unlike the GR case it's globally inertial. Since there's no gravity you could make your test sphere light years in size and it still wouldn't change with time.
The absence of gravity affects the rocket as well. With no gravity the rocket wouldn't hover above the Earth but instead it would go shooting off into outer space at an acceleration of 9.81 m/s$^2$. However inside the rocket you couldn't tell the difference. When you released your test articles they'd still fall to the floor in the same way, so your frame is still non-inertial.
Actually the fact that you couldn't tell the difference between gravity and the rocket accelerating away is a key part of general relativity, and it's called the equivalence principle.
This is an excellent question which is much more subtle than it first appears.
On a first (or even a second) reading, you seem to answer your own question and then ignore your own answer!
What I mean is, you clearly understand that while in an accelerating vehicle you experience a 'jerk' which does not happen with uniform motion, and you also acknowledge that the rest frame of the car is not inertial.
To an experienced physicist this already answers the question of why we declare acceleration to be absolute, but it does not satisfy you. Why? Because (as I realized on a third reading) your question is more interesting than that...
You don't care whether you are in an inertial frame or not. In your frame of reference you have the ability and the natural-born right to measure your own accelerations, and who are physicists to tell you that your answers are somehow wrong! Inertial reference frame or not, different frames can give different accelerations, and therefore acceleration is relative, right?
Well, guess what? I hear ya brother! You're absolutely right, and physicists need to talk more clearly.
Let's call a kinematic quantity (position, velocity, acceleration, or any higher derivatives) a Relative Quantity if its value is measured in arbitrary frame coordinates. So we have Relative Position, Relative Velocity, Relative Acceleration, etc.
The quantity called Absolute Acceleration (which we usually abbreviate to just 'Acceleration') is introduced as an additional quantity to the relative quantities, and for a very good reason I'll get to. Your 'Relative Acceleration' still exists, and yes, you have every right to use it, but you must also be aware of a much more interesting and fundamental quantity.
You see, whereas I have no experimental reason to justify the concept of an Absolute Position or an Absolute Velocity (i.e. they have no distinguishing features), I can point to an empirical justification for a particular acceleration.
Specifically, if I define Absolute Acceleration to be the Relative Acceleration measured within any inertial reference frame (they'll all agree on that measurement by the way), then for some mysterious reason of nature, Newton's Laws will work!
In particular, a body that is sufficiently isolated from all influences will have zero acceleration. When other stuff is in the vicinity, accelerations can be accounted for by well-defined force laws that 'make sense', given the nature of the surroundings. In particular, forces come in pairs (as per the third law), rather than popping up out of nowhere.
But this wonderful scheme only works out this way if we choose to measure accelerations relative to inertial reference frames, so we declare those accelerations to be 'true' or 'absolute', even though those adjectives are not as self-explanatory as physicists would have us believe.
That much answers your question, but the next bit is really worth knowing too...
When you use the framework of spacetime, rather than space + time, the notions of position, velocity, and acceleration, disappear, and you get a more satisfying geometric way to think about all this. Basically, spacetime distinguishes straight lines from curved lines, with the straight lines corresponding to uniform/inertial motion and the curved lines corresponding to non-inertial/accelerated motion.
There are simple experiments you can do to tell whether you are following a straight line or a curved line in spacetime. If you experience a 'jerk', you're on a curved line, for example.
Best Answer
The physics answer is, an accelerometer will detect all accelerations relative to an inertial frame. If you're in free fall being accelerated by a gravitational field, the answer is actually no, because a frame in free fall is inertial, even though for most purposes it's more useful to treat it as an accelerating frame.
So to your questions in order,
1a Yes, but free fall counts as inertial.
2a Only if you are also accelerating w/r/t an inertial frame.
1b No, if by "while I'm in motion" you mean constant velocity.
2b Yes, subject to the above.
The engineering/biology answer is flat "No." You specified that every part of you is being identically accelerated, and if that's the case, there's no way for an accelerometer (biological or otherwise) to be designed such that it will detect any acceleration. An accelerometer works by measuring the difference in motion between a mostly-inertial frame (like a mass on a spring, or fluid in your inner ear) and the accelerating frame (the body of the accelerometer, or your skull). If every particle is identically accelerating, there's nothing to measure.