Regularity of FEM a priori

Question

I am studying the book Finite Element Analysis by Szabó and Babuska, and specifically, section 1.5.1 on regularity defines $$ u_{EX} = x^{\alpha} \phi (x), \ \alpha > 1/2, \ x \in I = (0, \ell). \tag{1}\label{eqn1} $$ The authors states

For $u_{EX}$ to be in the energy space, its first derivative must be square integrable on $I$. Therefore $$ \int_0^\ell x^{2(\alpha-1)} dx > 0 \tag{2}\label{eqn2} $$ from which it follows that $\alpha$ must be greater than $1/2$.

My questions:

1. Might be trivial but how does condition \eqref{eqn2} follow from integrating $u'_{EX}= x^{\alpha-1} [\alpha \phi(x) + x \phi'(x)]$?
2. How does equation $\eqref{eqn1}$ relate to polynomial interpolating functions mostly used in practice?
3. More generally, how is $\alpha$, the convergence rate connected to the index of the sobolev space on which solutions live?

Thanks in advance for your help.