Using quantiative analysis to find empirical and molecular formula
The problem you gave doesn't quite work out, so I searched it up; did you perhaps mean $0.115 \space \text{g}$ of the sample were used in finding the sulfur content (you said $0.15 \space \text{g}$ instead).

If you did mean $0.115 \space \text{g}$, then we can figure out that the molecular formula of the unknown compound is $\text{CaC}_2\text{N}_2\text{S}_2$. Let me explain:

First, let's try figuring out how many moles of $\text{Ca}$ there are for each mole of the unknown compound, using the information that $0.250\space \text{g}$ of the compound was used to form $0.16 \space \text{g}$ of CaCO3. Converting everything to moles, this means we used $(0.250\space \text{g})\space/\space (156 \space \text{g/mol})=0.00160 \space \text{mol}$ of the unknown compound to get $(0.16\space \text{g})\space/\space (100.0869 \space \text{g/mol})=0.00160\space \text{mol CaCO}_3$, or $0.00160\space \text{mol Ca}$. This means that one mole of the unknown compound will contain $(0.00160\space \text{mol Ca})\space/\space (0.00160 \space \text{mol Unknown Compound}) \space\times\space (1 \space\text{mol Unknown Compound})=1\space\text{mol Ca}$

We can do this for each of the other procedures to get that for each mole of the unknown substance, we have $1\space\text{mol Ca}$, $2\space\text{mol S}$, and $2\space\text{mol N}$. Now, we can just subtract the masses of $\text{Ca}$, $\text{S}$, and $\text{N}$ from the molar mass of the unknown compound to get the mass of $\text{C}$ in the compound. For each mole of the unknown compound, we get that we have $\text{156 g}-\text{40.08 g}-2(\text{14.01 g})-2(\text{32.06 g})=23.8 \space\text{g}$ of carbon. This is just $(23.8\space \text{g})\space/\space (12.01 \space \text{g/mol})=1.98\space \text{mol} \approx 2 \space \text{mol}$ carbon per mole of unknown compound.
So, since we have $1\space\text{mol Ca}$, $2\space\text{mol S}$,  $2\space\text{mol N}$, and $2\space\text{mol C}$ for each mole of the unknown compound, the unknown compound must be $\boxed{\text{CaC}_2\text{N}_2\text{S}_2}$.