Step 1: Write the reactions of borax.
On hydrolysis,
\[
\mathrm{Na_2B_4O_7+7H_2O
\rightarrow
2NaOH+4H_3BO_3.}
\]
Thus,
\[
A=\mathrm{H_3BO_3},
\qquad
B=\mathrm{NaOH}.
\]
In the presence of hydrochloric acid,
\[
\mathrm{Na_2B_4O_7+2HCl+5H_2O
\rightarrow
2NaCl+4H_3BO_3.}
\]
Hence,
\[
C=\mathrm{H_3BO_3}.
\]
Step 2: Check the given statements.
\[
A=\mathrm{H_3BO_3}
\]
is a weak acid, not a strong base.
Therefore,
Statement I is incorrect.
\[
B=\mathrm{NaOH}
\]
is a white crystalline solid having a soapy touch.
Hence,
Statement II is correct.
Boric acid,
\[
\mathrm{H_3BO_3},
\]
is soluble in water.
Thus,
Statement III is also correct.
Step 3: Choose the correct option.
Therefore,
\[
\boxed{\text{Only Statements II and III are correct}.}
\]
Hence, the correct option is \(\boxed{(B)}\).