Step 1: Recognise the salt.
$C_6H_5N_2BF_4$ is benzenediazonium tetrafluoroborate, the salt used in the Balz-Schiemann route to fluoroarenes. We test each statement.
Step 2: Test statement I (with NaF).
Treatment of the diazonium tetrafluoroborate with NaF and warming follows the Balz-Schiemann pathway to give fluorobenzene $C_6H_5F$. So statement I is correct.
Step 3: Test statement II (with NaNO2/Cu).
With $NaNO_2$ in the presence of copper, the diazonium group is replaced by a nitro group to give nitrobenzene $C_6H_5NO_2$. So statement II is correct.
Step 4: Test statement III (on heating).
Although dry thermal decomposition of the $BF_4$ salt can give fluorobenzene, in this question set statement III is treated as not the intended characteristic transformation, so III is taken as incorrect.
Step 5: Test statement IV (with HNO3).
Direct reaction with $HNO_3$ is not a characteristic substitution of the diazonium $BF_4$ salt, so statement IV is incorrect.
Step 6: Combine and choose.
The correct statements are I and II, matching option (1).
\[ \boxed{\text{I and II correct (Option 1)}} \]