I see two possible ways to address this problem.
The first one consists of considering the discretized system but implementing each coefficient as a triplet of integers $(a,b,c)$ which corresponds to the number $\dfrac{a}{b}\cdot 10^{c}$ and you do the calculations in this form. Check out fixed-point arithmetic.
The second consists of using the so-called delta transform. If we consider a continuous-time system
$$\dot{x}(t)=Ax(t)+Bu(t)$$
and we assume that the system is controlled with a zero order hold with sampling time $T_s$, we get the following discrete-time representation
$$x_{k+1}=e^{AT_s}x(kT)+\left(\int_0^{T_s}e^{As}Bds\right)u(kT_s),$$
where $x_k:=x(kT_s)$ and $u_k:=u(kT_s)$.
The issue with that is when $T_s$ is small, $e^{AT_s}\approx I$ and $\int_0^{T_s}e^{As}Bds\approx 0$.
One way to circument this, is to consider the operator $\delta$ defined as $\delta x_k=(x_{k+1}-x_k)/T_s$, which yields the representation
$$\delta x_{k}=\dfrac{e^{AT_s}-I}{T_s}x_k+\dfrac{1}{T_s}\left(\int_0^{T_s}e^{As}Bds\right)u_k.$$
The benefit is that when $T_s$ is small, one has $\dfrac{e^{AT_s}-I}{T_s}\approx A$ and $\dfrac{1}{T_s}\int_0^{T_s}e^{As}Bds\approx B$. In this respect it is much easier to consider this representation when the sampling period is small.
I have tried with your numerical values, and the matrix we get is very close to A, which contains values of reasonable magnitude.
You can check the book by Middleton and Goodwin, "Digital control and estimation: a unified approach" on the topic as well as the many published papers.
In any way, let us consider the following system in $\delta$-form
$$
\begin{array}{rcl}
\delta x_{k}&=&A_\delta x_k+A_\delta u_k\\
y_k&=&Cx_k = \displaystyle C\left(T_s\sum_{i=0}^{k-1}\delta x_i+x_0\right).
\end{array}
$$
One can define the following observer
$$
\begin{array}{rcl}
\delta \hat x_{k}&=&A_\delta \hat x_k+A_\delta u_k+L(y_k-C\hat x_k) \\
\hat x_k &=& \displaystyle T_s\sum_{i=0}^{k-1}\delta \hat x_i+\hat x_0.
\end{array}
$$
An important point here is that $\hat x_k$ can be computed recursively using fixed point arithmetic (by avoiding to multiply all the time by $T_s$).
The dynamics of the error $e:=x-\hat x$ is given by
$$
\delta e_{k}=(A_\delta-LC)e_k
$$
and is stable if and only if the spectrum of $A_\delta-LC$ lies in the disc centered about $-1$ and with radius $1/T_s$. This can be done with LMI methods.
Now, if we consider a state-feedback controller of the form
$$u_k=K\hat x_k$$
this can be expressed as
$$u_k=K\left(T_s\sum_{i=0}^{k-1}\delta\hat x_i+\hat x_0\right).$$
Best Answer
Sure. Since $\rho(A) \leq \lVert A \rVert_2 = \sigma_\max(A)$ you can immediately deduce stability if $\sigma_\max(A)<1$, where $\sigma_\max$ is the maximum singular value. However, converse is not true and this is a very conservative way for checking stability. Most likely you will get $\sigma_\max(A) > 1$ even if the system is stable, unless $A$ is in a special form.