(A) given an example of sequence $f_n:[0,1] \to \mathbb{R}$ such that $\{f_n\}^\infty_{n=1} $ are continuous, uniformly bounded, converging pointwise without having a uniformly convergent subsequence.
(B) Can the $\{f_n\}^\infty_{n=1} $ in the above example be an equicontinuous family?
$f_n:[0,1] \to \mathbb{R}$ defined by $f_n(x)=x^n$, does it satisfy all the above properties ?
Yes, your example satisfies all the properties that you mentioned: each $f_n$ is continuous, the sequence is uniformly bounded (the range of each $f_n$ is a subset of $[0,1]$) and it has no uniformly convergent subsequence (such a subsequence would converge uniformly to a discontinuous function, which is impossible).