This is a classic example of the kinetics of reversible reactions.
From the mechanism $\ce{A <=>[$k_\mathrm f$][$k_\mathrm b$] B}$ you can write the kinetic equations
\begin{align}
\frac{\mathrm d[\ce{A}]}{\mathrm dt} &= - k_a [\ce{A}] + k_b [\ce{B}]\\
\frac{\mathrm d[\ce{B}]}{\mathrm dt} & = k_a [\ce{A}] - k_b [\ce{B}]\\
\end{align}
To obtain the dependence of the concentration with time, you must solve this set of differential equations, but its solution is
trivial since both concentrations are not independent
\begin{equation}
[\ce{A}]_0 + [\ce{B}]_0 = constant = [\ce{A}]_\mathrm{t} + [\ce{B}]_\mathrm{t}
\end{equation}
thus
\begin{equation}
[\ce{B}]_\mathrm{t} = constant - [\ce{A}]_\mathrm{t}
\end{equation}
In the case you describe $[\ce{B}]_0 = 0$ so
\begin{equation}
[\ce{B}]_\mathrm{t} = [\ce{A}]_0 - [\ce{A}]_\mathrm{t}
\end{equation}
If the initial concentration of the species A is $[\ce{A}]_0 = a$, and $[\ce{B}]_\mathrm{t} = x$, $\mathrm{[A]_t = a -x} $
\begin{equation}
\mathrm{ \frac{d[\ce{A}]}{dt} = -\frac{d x}{dt} = - k_a (a-x) + k_b x}
\end{equation}
The solution of this differential equation is simple and the result is
\begin{equation}
\mathrm{ x_{eq} - x = x_{eq} \, e^{-(k_a + k_b) t}}
\end{equation}
where $\mathrm{x_{eq} = |B|_{eq}}$.
You write this expression as a function of $\mathrm{|A|_t}$ and $\mathrm{|B|_t}$ as
\begin{align}
\mathrm{|A|_t \,=\,} & \mathrm{|A|_{eq} + |B|_{eq}e^{-(k_a + k_b) t}}\\
\mathrm{|B|_t \,=\,} & \mathrm{|B|_{eq} (1 -e^{-(k_a + k_b) t})}\\
\end{align}
Thus this reaction behaves as a first order reaction with $\mathrm{k_1 = k_a + k_b}$ and with $\mathrm{|A|_\infty = |A|_{eq}}$ and $\mathrm{|B|_\infty = |B|_{eq}}$


Furthermore, once the reaction has reached the equilibrium
\begin{equation}
\mathrm{v_a} \,=\, \mathrm{v_b}
\end{equation}
\begin{equation}
\mathrm{k_a\,|A|_{eq}} \,=\, \mathrm{k_b\,|B|_{eq} }
\end{equation}
so
\begin{equation}
\mathrm{K_{eq}} \,=\, \frac{\mathrm{|B|_{eq}}}{\mathrm{|A|_{eq}}} \,=\, \frac{\mathrm{k_a}}{\mathrm{k_b}}
\end{equation}
Using the equilibrium equations and the integrated rate equations you can answer the questions.
Bibliography.
This particular mechanism is a classic in chemical kinetics.
You can find more information on general physical chemistry or introductory chemical kinetics textbooks. For example:
T. Engel et al., Physical Chemistry, 3rd Edition, Pearson (2013)
M. J. Pilling and P. W. Seakins, Reaction Kinetics, 2nd Edition, Oxford University Press (1996)
K. J. Laidler, Chemical Kinetic, 3rd edition, Pearson (1997)