Question: Can the following three conjectured closed form representations be proven to be correct assuming $\Re(s)>1$ and $m\in \mathbb{P}$?
$$\underset{N\to\infty}{\text{lim}}\left(-\sum\limits_{n=1}^N a(n) \frac{n\, m^{n\, s}}{\left(m^{n\, s}-1\right)^2}\right)=\frac{1}{1-m^s}\tag{1}$$
$$\underset{N\to\infty}{\text{lim}}\left(\sum_{n=1}^N \frac{a(n)}{m^{n s}-1}\right)=\log\left(\frac{1}{1-m^{-s}}\right)\tag{2}$$
$$\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N \frac{\phi(n)}{m^{n s}-1}\right)=\frac{1}{m^s+m^{-s}-2}\tag{3}$$
The arithmetic function $a(n)$ referenced on the left sides of formulas (1) and (2) above is defined as
$$a(n)=\frac{1}{n}\sum\limits_{d|n} \mu(d)\, d\tag{4}$$
and $\phi(n)$ referenced on the left side of formula (3) above is Euler's totient function.
Note that $a(n)\, n=\sum\limits_{d|n} \mu(d)\, d$ corresponds to OEIS entry A023900 Dirichlet inverse of Euler totient function (A000010) and
$$\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N A023900(n)\, n^{-s}\right)=\frac{\zeta(s)}{\zeta(s-1)},\quad\Re(s)\ge 2\tag{5}$$
whereas
$$\underset{\text{N}\to\infty}{\text{lim}}\left(\sum_{n=1}^N \phi(n)\, n^{-s}\right)=\frac{\zeta(s-1)}{\zeta(s)},\quad\Re(s)>2\tag{6}$$
I suspect the closed-form representations in formulas (1) to (3) above may actually be more generally valid for $\Re(s)>0$ and integer $m>1$ or perhaps even $m>1$, but my primary interest is the more limited case stated in the question above for reasons that will become apparent in the formulas below.
The left and right sides of formula (1) above are related to the following two formulas for $\frac{\zeta'(s)}{\zeta(s)}$
$$\frac{\zeta'(s)}{\zeta(s)}=\underset{M,N\to\infty}{\text{lim}}\left(-\sum\limits_{m=2}^M 1_{m\in\mathbb{P}}\, \log(m) \sum\limits_{n=1}^N a(n) \frac{n\, m^{n\, s}}{\left(m^{n\, s}-1\right)^2}\right),\quad\Re(s)>1\tag{7}$$
$$\frac{\zeta'(s)}{\zeta(s)}=\underset{M\to\infty}{\text{lim}}\left(\sum\limits_{m=2}^M 1_{m\in\mathbb{P}}\, \frac{\log(m)}{1-m^s}\right),\quad\Re(s)>1\tag{8}$$
where $1_{m\in\mathbb{P}}$ is the prime indicator function which returns the value $1$ if $m$ is a prime and zero otherwise.
The left and right sides of formula (2) above are related to the following two formulas for $\log\zeta(s)$:
$$\log\zeta(s)=\underset{M,N\to\infty}{\text{lim}}\left(\sum\limits_{m=2}^M 1_{m\in\mathbb{P}} \sum _{n=1}^N \frac{a(n)}{m^{n s}-1}\right),\quad\Re(s)>1\tag{9}$$
$$\log\zeta(s)=\underset{M\to\infty}{\text{lim}}\left(\sum\limits_{m=2}^M 1_{m\in\mathbb{P}}\, \log\left(\frac{1}{1-m^{-s}}\right)\right),\quad\Re(s)>1\tag{10}$$
The left and right sides of formula (3) above are related to the following two formulas for $K_\Omega(s)$
$$K_\Omega(s)=\underset{M,N\to\infty}{\text{lim}}\left(\sum\limits_{m=2}^M 1_{m\in\mathbb{P}}\, \sum\limits_{n=1}^N \frac{\phi(n)}{m^{n s}-1}\right),\quad\Re(s)>1\tag{11}$$
$$K_\Omega(s)=\underset{M\to\infty}{\text{lim}}\left(\sum\limits_{m=2}^M \frac{1_{m\in\mathbb{P}}}{m^s+m^{-s}-2}\right),\quad\Re(s)>1\tag{12}$$
which represent alternate ways to evaluate the Dirichlet series
$$K_\Omega(s)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N 1_{n=p^k}\, \Omega(n)\, n^{-s}\right),\quad\Re(s)>1\tag{13}$$
where $1_{n=p^k}$ is the prime-power indicator function which returns the value $1$ if $n$ is a prime-power and zero otherwise and $\Omega(n)$ counts the number of non-distinct primes in $n$ (i.e. $\Omega(n)=k$ when $n=p^k$).
I believe formula (13) for $K_\Omega(s)$ above can be analytically continued from $\Re(s)>1$ to $\Re(s)>0$ as follows
$$K_\Omega(s)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N b(n) \log(\zeta(n s))\right),\quad\Re(s)>0\tag{14}$$
where
$$b(n)=\frac{1}{n}\sum\limits_{d|n} d^2 \mu\left(\frac{n}{d}\right)\tag{15}$$
Note that $b(n)\, n=\sum\limits_{d|n} d^2 \mu\left(\frac{n}{d}\right)$ corresponds to OEIS Entry A007434 for the Jordan function $J_2(n)$ which is a generalization of Euler's totient function $\phi(n)$ and
$$\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N \left(\sum\limits_{d|n} d^2 \mu\left(\frac{n}{d}\right)\right) n^{-s}\right)=\frac{\zeta(s-2)}{\zeta(s)},\quad\Re(s)>3\tag{16}$$
I believe formulas (7) to (12) above for $\frac{\zeta'(s)}{\zeta(s)}$, $\log\zeta(s)$, and $K_\Omega(s)$ are all valid for $\Re(s)>1$, but note formula (12) above for $K_\Omega(s)$ converges for $\Re(s)<-1$ as well as $\Re(s)>1$ and evaluates to $K_\Omega(-s)$=$K_\Omega(s)$ for $|\Re(s)|>1$.
For simplicity and generality make the definition $\,z := m^{-s}.\,$ Then your three equations become $$ -\sum_{n=1}^\infty a(n) \frac{n\,z^{-n}}{(z^{-n}-1)^2} =\frac{1}{1-z^{-1}},\tag{1}$$ $$ \sum_{n=1}^\infty \frac{a(n)}{z^{-n}-1} = \log\Big(\frac{1}{1-z}\Big), \tag{2}$$ $$ \sum_{n=1}^\infty \frac{\phi(n)}{z^{-n}-1} = \frac{z}{(1-z)^2}. \tag{3}$$ They all have radius of convergence $1$. For some background on series such as these refer to Wikipedia article Lambert series.