The experiments you describe can all be analyzed in a flat spacetime using SR. We can switch back and forth between an inertial frame A and an accelerated frame B. The equivalence principle says that observers in B see a gravitational field. (That is, we have a gravitational field and a varying gravitational potential, but spacetime is flat.)
Observer Alice in inertial frame A describes the source and receiver as accelerating, so she interprets the observations as a kinematic Doppler effect. Bob in B sees source and receiver as both being at rest, so he interprets the effect as gravitational rather than kinematic. He can call it a gravitational Doppler shift, or he can call it a gravitational time dilation. GR doesn't distinguish between the two.
If you wanted to construct a theory in which the distinction between gravitational Doppler shifts and gravitational time dilation was meaningful, an example of how you could do that would be that your theory could predict that clocks of different types that were rate-matched in one location could become mismatched in rate when moved together to some other gravitational potential. This would clearly be an example of gravitational time dilation, not a Doppler shift, because arguments about Doppler shifts can't explain the difference in behavior between the two clocks. But this theory is not a metric theory and it violates the equivalence principle (because Alice sees no gravitational field and therefore can't explain what's going on). Because GR is a metric theory that incorporates the equivalence principle, the distinction you're trying to make isn't a distinction that GR can make.
So the result is that you can call this effect a kinematic Doppler effect, a gravitational Doppler effect, or gravitational time dilation, and all of these interpretations are equally valid.
The difficultly you have imaging it could be because you imagine time like a series of panels in a cartoon and wonder if you are missing a foot in some panel.
But better to imagine a 4d space that labels events. An event is a combination of a place and a time. For instance the event when-where a light goes on, or the when-where a section of DNA unzips so that it can replicate.
The universe is made of such events. And they don't particularly care what labels you assign them. They don't care if you call an event origin or whether you labeled it as later or before the origin or if you labeled it as being in the x direction of the origin or in the y direction of the origin. Perhaps there is no natural path between one event and the other.
What you need is that you have events and that they affect each other. Take the DNA unzipping. It does this because molecules around it get close enough to make it do so. How do they get close? Firstly the parts to make it could be nearby or it could be part of something else nearby or it could be moving towards that area and inertia carries it on.
If it is inertia then it arrives when-where it does because of the speed it had. If it is something larger falling apart then it depends on the rate things fall apart (which depends on the things around it) and if it forms then it depends on the rate thibgs form (which depends on the things around it). But no matter what the things start to happen at one event (at one when-where) and finish at another event (another when-where) and while the labels don't matter that curve with a fixed endpoint and a fixed starting point and a whole path on 4d spacetime it has a length. And that length is determined by the geometry. So that is where gravity comes in. Those rates are measured against the 4d length of that curve.
And that curves length isn't the length it looks like in a picture it is the one given by the equations.
Which means there is a nice way to visualize it. Imagine that everywhere along spacetime there are all these curves. And that things won't do their things until they get paid money. And imagine that the money they are paid depends on this length, not the visual length, but the one given by the equations.
So its like you pay people less when deeper in gravity well so they do everything less quickly. Radioactive particles decay less quickly. Cells repair themselves from radioactive damage less quickly. Brains register pain less quickly. Signals travel less quickly. Because every measure of rates is affected by the lengths if these curves.
So your foot feels like signals from your head are arriving more often than they do because the foot is acting slower. And the signals from the foot to the head seem to arrive less often than the do because the head does things faster.
With this idea of everything waiting to get paid you can see that we have no idea who is getting paid the normal rate and who is underpaid. We just know that each one only does things when it has been paid enough and that curves inside you foot and curves inside your head have to measure based on the geometry then.
So you head feels like there are more breaths in your lungs per thing it does and you for feels like there is less. Same with beats of tour heart. They might feels like you are eating less or more often than it does other things.
But that just means the feedback they give about whether they are getting oxygen compared to how much they are using is based on that. Blood can circulate and those cells that need it most can get ready to grab it and if the amount coming back it too depleted that can signal your lungs to breath more often. You lungs don't have to keep track of it was because you head was working harder and so more cells there needed oxygen or whether it was because you stood up and so your feet were now in a deeper well.
All the things that affect your body affect your body not just the rates of time. Just like if you paid different regions different amounts or they had less or more holidays. So your feet take less oxygen because they do everything more slowly. But they also multiply more slowly, repair more slowly, and so on. Since the rest a bit slower your brain when learning to control them it learned about that, just as it learned about the exact amount of time it takes to send a signal through the specific paths ways from your brain to your foot.
Just as you'd learn if another country had way more holidays you'd learn to send packages sooner. But this is a small effect compared to a cell just being farther away and your body is adaptable.
So you deal. You just have to accept that every single thing happens slower the good and the bad. And these effects will be totally swamped by bugger effects like a bacteria getting in the way. So you won't notice.
Mostly I tried to address the cognitive bias that might have been making it difficult. Every last of your body is someplace at every time, but the rate that they react to things depends on the path they take.
There are other similarly small effects. There is a Doppler broadening of binding energies because of the special relativistic time dilation from the motion. Bit that is covered in what I said, the special relativity notion requires that it move through spacetime so the length of the path takes the notion and the gravity well into account.
So increasing the temperature of your body also makes it age slower, but when that happens it can also break things by making them hit harder against each other.
Best Answer
The time dilation is due to a difference in the gravitational potential energy, so it is due to the difference in height. It doesn't matter whether the strength of the gravitational field varies, or how much it varies, all that matters is that the two observers comparing their clocks have a different gravitational potential energy.
To be more precise about this, when the gravitational fields are relatively weak (which basically means everywhere well away from a black hole) we can use an approximation to general relativity called the weak field limit. In this case the relative time dilation of two observers $A$ and $B$ is given by:
$$ \frac{dt_A}{dt_B} \approx \sqrt{ 1 + \frac{2\Delta\phi_{AB}}{c^2}} \approx 1 + \frac{\Delta\phi_{AB}}{c^2} \tag{1}$$
where $\Delta\phi_{AB}$ is the difference in the gravitational potential energy per unit mass between $A$ and $B$.
Suppose the distance in height between the two observers is $h$, then in a constant gravitatioinal field with acceleration $g$ we'd have:
$$ \Delta\phi_{AB} = gh $$
If this was on the Earth then taking into account the change in the gravitational potential energy with height we'd have:
$$ \Delta\phi_{AB} = \frac{GM}{r_A} - \frac{GM}{r_B} $$
where $r_A$ and $r_B$ are the distances of $A$ and $B$ from the centre of the earth and $M$ is the mass of the Earth. Either way when we substitute our value of $\Delta\phi_{AB}$ into equation (1) we're going to get a time dilation.
As for the accelerating rocket: the shortcut is to appeal to the equivalence principle. If acceleration is equivalent to a gravitational field then it must also cause a time dilation in the same way that a gravitational field does.
Alternatively we can do the calculation rigorously. The spacetime geometry of an accelerating frame is described by the Rindler metric, and we can use this to calculate the time dilation. The Rindler metric for an acceleration $g$ in the $x$ direction is:
$$ c^2d\tau^2 = \left(1 + \frac{gx}{c^2}\right)^2c^2dt^2 - dx^2 - dy^2 - dz^2 $$
We get the time dilation by setting $dx=dy=dz=0$ to give:
$$ c^2d\tau^2 = \left(1 + \frac{gx}{c^2}\right)^2c^2dt^2 $$
and on rearranging this gives:
$$ \frac{d\tau}{dt} = 1 + \frac{gx}{c^2} $$
which is just the equation (1) that we started with.