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: 
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> Can mathematics of relativity be interpreted in such a way that time is invariant and everything else is flexible?
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Not to my knowledge, no. The time you call “flexible” is coordinate time and is frame variant and cannot be made invariant. However, in relativity there is also proper time, which is invariant. This is the time measured by a clock. 

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> time is such a complex concept that we cannot measure it
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This is phenomenally wrong. Not only can we measure time, we can measure it far more precisely than any other quantity. A large part of modern metrology is converting other measurements into measurements of time specifically so that we can use our ability to precisely measure time to precisely measure other quantities. 

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> If not, what is the problem with that and why speed of light is given that much of importance?
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The speed of light is important because it gives the conversion between our units of time and space. Once it is recognized that time and space belong together in a single geometric structure (spacetime) then many previously separate things become connected.
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> Can the mathematics of relativity be interpreted in such a way that time is invariant and everything else is flexible?
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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)