Step 1: Check statement I.
Liquid sodium has very high thermal conductivity and carries heat efficiently, so it is genuinely used as a coolant in fast breeder nuclear reactors. Statement I is correct.
Step 2: Recall the meaning of deliquescent.
A deliquescent solid absorbs so much atmospheric moisture that it dissolves in the absorbed water.
Step 3: Check statement II.
The small, highly polarizing $\text{Li}^+$ ion makes LiCl strongly moisture absorbing, so LiCl is deliquescent. Statement II is correct.
Step 4: Recall the solubility balance.
Solubility depends on lattice energy versus hydration energy; when one ion is tiny or both ions are large, solubility drops.
Step 5: Check statement III.
LiF has a very high lattice energy from two small ions, while CsI has low hydration energy from two large ions, so both are poorly soluble. Statement III is correct.
Step 6: Combine.
All three statements hold, which is option 4.
\[ \boxed{\text{I, II, III}} \]