Gravitational time dilation caused by a galaxy, and by In a word, if you are sitting on the Earth, if I'm not mistaken you are experiencing Time Dilation compared to being in deep solar system space. Due to the mass of the Earth.
However. We're all sitting in a galaxy, and it weighs a lot.
(Perhaps 2e40 kg? I have no idea if issues like "dark matter" radically effect this.)
Can anyone quantify,
a) my Time Dilation sitting on Earth - versus in deeper space
b) my Time Dilation sitting in the Milky Way - versus deeper in space
But wait.
We're all in the Universe.  It weighs a lot.
In fact ..... are we all experiencing Time Dilation because of this being-in-the-rather-heavy-Universe affair?
c) if so, how much?  Thanks.
Important ancillary question a'): I've never quite found the answer to this: we experience Time Dilation due to planet Earth. Now, if you are at the center of the Earth, you experience no gravitational pull, but, do you experience the Time Dilation??
 A: In the weak field limit, which applies to all the cases you've described, the difference between the time rates for two observers with a Newtonian gravitational potential energy difference of $\Delta\Phi$ is given by:
$$ \frac{\Delta t_1}{\Delta t_2} = \sqrt{1 - \frac{2\Delta\Phi}{c^2}} \tag{1} $$
Note that the time dilation is related to the gravitational potential energy not the gravitational force - you'll see why this matters when we come to your ancillary question.
So to answer your questions (a) and (b) just work out what the difference in gravitational potential energy between your two observers and plug it into equation (1). I'll leave this as an exercise for the reader.
The answer to question (c) is a bit subtle, because the key feature of an FLRW universe is that it is homogenous i.e. the gravitational potential is the same everywhere in the universe. That means whatever pair of observers you choose $\Delta\Phi$ is always zero and therefore the time dilation is always zero. You can't ask about the time dilation relative to an observer outside the universe because an FLRW universe has no outside.
Now on to the ancilliary question: at the centre of the Earth the gravitational force is indeed zero, but the gravitational potential is not. As you move from infinity to the surface of the earth $\Phi(r)$ decreases (i.e. gets more negative) as $r^{-1}$, but as you move below the surface to the centre $\Phi(r)$ carries on decreasing only not as fast. So compared to an observer at infinity the time dilation at the centre of the Earth is greater than the time dilation at the surface.
Actually I've just spotted this has already been addressed in the question Gravitational time dilation at the earth's center.
A: It doesn't make sense to say "am I experiencing time dialation?"
It only makes sense to compare two different observers, and ask whether one of them observes the others' clock to be ticking more slowly, say, when they are looking at minimum distance light rays coming from the other observer.  
With this in mind, the answers to most of your questions are "yes", "it's hard to measure", and "the question is ill-defined."
As for comparing to "deeper space", you have to figure out exactly what you mean by "deeper space".  Galaxies are closer together, relative to the size of the galaxies, than planets and stars are.  Get too far from the Milky way, and you're already under the influence of Andromeda.  In fact, look at how big Andromeda would be in the night sky, if it wasn't outshone by nearer objects: 
http://www.dmuller.net/cosmology/andromeda.php
