I have the following question:
I'm a bit confused as to why the expected time would not contain a "+a" term. The way I did it was to first calculate the expected number of customers (Y) to arrive: $$E[Y]= (1 + E[Y])(1 - e^{-\lambda a})$$ So we get that the expected number of customers is $e^{\lambda a} -1$. Then, we can just multiply by the expected time for each customer to arrive, and then add $a$ since the shop will be kept open for $a$ time after the last customer.
Where's the flaw in my reasoning here?
Your approach doesn’t work because the customers that do arrive take less than average time to arrive; their mean arrival time is just
\begin{eqnarray} \frac{\int_0^at\mathrm e^{-\lambda t}\mathrm dt}{\int_0^a\mathrm e^{-\lambda t}\mathrm dt} &=& \frac1\lambda\cdot\frac{1-\mathrm e^{-\lambda a}(1+\lambda a)}{1-\mathrm e^{-\lambda a}} \\ &=& \frac1\lambda-\frac a{\mathrm e^{\lambda a}-1}\;. \end{eqnarray}
The second term cancels the term that you were missing.
Alternatively, you can do the first-step analysis that you did for the number of customers for the opening time $X$:
\begin{eqnarray} E[X] &=& a\mathrm e^{-\lambda a}+\lambda\int_0^a(t+E[X])\mathrm e^{-\lambda t}\mathrm dt \\ &=& a\mathrm e^{-\lambda a}+\frac1\lambda\left(1-\mathrm e^{-\lambda a}(1+\lambda a)\right)+E[X]\left(1-\mathrm e^{-\lambda a}\right)\;, \end{eqnarray}
and again the term you were missing cancels and the solution is
$$ E[X]=\frac{\mathrm e^{\lambda a}-1}\lambda\;. $$
Another check on the solution is that the limit for $\lambda\to0$ should be $a$, not $2a$.