Time, its measurement, frames Hartle, gravity pg 34

" An observer in a inertial frame can construct a clock that measures the time t"

In physics is time something else that exists irrespective of clocks? What we do is just that we measure it?
Pg 35

" ...It's a central assumption of Newtonian mechanics that there is a single notion of time for all the inertial observers"

What does the above sentence exactly mean?
Does it mean that whatever time is, it's same for all inertial frames? So if I take similarly made contraptions that serve as clocks, to every inertial frame,they would all tick at same rate with respect to one another?
What about non inertial frames? Why were non inertial frames excluded from

"...that there is a single notion of time for all the inertial observers".

 A: 
In physics is time something else that exists irrespective of clocks?
What we do is just that we measure it?

I think the term "clock" is a bit ambiguous, but there are things we can say about them.
In the statement you refer to, it is implied that the "clock" is a manufactured clock used by humans to measure the passage of time. In other contexts, "clock" can mean the rate at which time passes for a thing in its own reference frame. For example, a radioactive element has a "clock" that determines how fast it will decay (on average); chemical reagents have a "clock" determining how fast they react; the atoms in springs have a "clock" determining how fast they vibrate.
If you wanted to, you could use these internal clocks to build a real clock (and people do).
The important thing is that all of these "clocks" measure the same abstract underlying rate of passage of time in the same reference frame.
Does that abstract rate of time exist independently of all clocks? Saying that there are no clocks is saying that there is nothing that experiences the passage of time, in which case it is probably meaningless to say whether time exists or not.
You can imagine that there is a reference frame moving through the universe at some speed that has no particles in it. Does time exist for it? It doesn't really matter whether it does or not.

What does the above sentence exactly mean?

In Newtonian physics, it is assumed that all clocks in the universe "tick" at the same rate. A piece of uranium in one place moving at one speed will decay at the same rate as another piece elsewhere moving at a different rate. We now know this is not true and that time can pass at different rates. Each thing can have its own clock.

What about non inertial frames? Why were non inertial frames excluded from

The mathematics of Special Relativity cover the special case of inertial (non accelerating) reference frames. For reference frames under acceleration (for example, in a gravitational field) that is the domain of General Relativity.
