Question: In relativity theory it is accepted that speed of light is invariant and even time is flexible. Can the mathematics of relativity be interpreted in such a way that time is invariant and everything else is flexible? Because time is such a complex concept that we cannot measure it or even theoretically define it, I think. If not, what is the problem with that and why is the speed of light given that much importance?

Answer: 
It is possible to measure time by counting reliably periodic events, such as the spinning of the Earth about its axis or the swings of a pendulum. Generally speaking, more accurate measurements can be made by picking events with shorter and more uniform periods. The most accurate clocks currently, known as atomic clocks, effectively count electron transitions between energy levels. The second is defined as the time which elapses over 9,192,631,770 of a particular type of transition in caesium atoms.

It used to be thought that time was the same everywhere- that was the assumption underlying Newtonian mechanics.

We now know from a lot of experimental evidence that the speed of light (in a vacuum) is a natural constant. One direct consequence of that is that the elapsed time between events will vary between moving reference frames. 

There is no problem with defining time theoretically as a dimension. Disagreements about the meaning of time arise in connection with questions we cannot address experimentally. For example, some people take the view that there is only the present; others take the view that the past continues to exist, and some take the view that the future already exists. If you are interested in those debates try the search string 'philosophy of time' in your favourite search engine.
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**No** : in-variance of $c$ is proved by [Kennedy & Thorndike](https://en.wikipedia.org/wiki/Kennedy%E2%80%93Thorndike_experiment) and variance of $t$ is proved by [Ives & Stilwell](https://en.wikipedia.org/wiki/Ives%E2%80%93Stilwell_experiment)