1) Does that mean that gravity exerts/draws energy on/off cesium thus making it decompose faster/slower?
I have no idea what you mean by "exerts/draws energy on/off". And atomic clocks don't measure cesium decomposing. But yes, the "periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom" - like everything else - will be slower in a gravitational well compared to things outside that well, as a result of gravitational time dilation.
2) I.e. does gravity influence the half-life of elements?
No, because of the way half-life is defined and measured. The element still decays at exactly the same rate relative to a watch sitting next to it, or the radiation corresponding to ... of a cesium 133 atom sitting next to it, so the decay will take the same number of seconds. But if they're in a gravitational well, then it will happen more slowly than a lump of the same element outside the gravitational well.
But if you're in the gravitational well, you won't observe that your atoms are decaying slowly - you'll observe that your ones are decaying normally and the other ones are decaying quickly. If you're outside it, you'll observe that your atoms are decaying normally and the other ones are decaying slowly.
Is the duration or amount of seconds elapsed in space different? Or both?
Yes. If you're standing in an extremely strong gravitational field (and somehow avoid being crushed), and look at a clock that isn't, you might see that 61 seconds pass on that clock for every 60 seconds on yours.