The Newton laws of gravity, on the way to establishing the Kepler laws, give a parametrization
$$
r=\frac{R}{1+e\cos(φ-ψ)} ~~ \text{ and } ~~ \dot r=\sqrt{\frac{GM_E}{R}}e\sin(φ-ψ),~~r^2\dot\phi = -\sqrt{RGM_E} .
$$
for the polar coordinates $(r(t),\phi(t))$, $x+iy=re^{i\phi}$, of the satellite (moving clock-wise if moving horizontally to the right). $ψ$ is the angle for the smallest radius, at periapsis, $\frac{R}{1+e}$.
The initial data in polar coordinates can be obtained from the given initial data that can easily be applied to Cartesian coordinates. As $x_0+iy_0=r_0e^{iφ_0}$, and the interpretation of the launch conditions as $x_0=0$, $y_0=R_E+H_{sat}$ gives $r_0=y_0$ and $φ_0=0$. The constants in the reduced equations follow from
$$
\frac{\dot x_0+i\dot y_0}{x_0+iy_0}=\frac{\dot r_0}{r_0}+i\dotφ_0
$$
with $\dot x_0=V_{sat}\cos\theta$, $\dot y_0=V_{sat}\sin\theta$, so that
$$
\dot r_0=\dot y_0=V_{sat}\sin\theta, ~~~ \dot φ_0=-\frac{\dot x_0}{y_0}=-\frac{V_{sat}}{r_0}\cos\theta
$$
Then
$$
r_0V_{sat}\cosθ=\sqrt{RGM_E}\implies R=\frac{r_0^2V_{sat}^2\cos^2θ}{GM_E}\\
e\cosψ=\frac{R}{r_0}-1 ~\text{ and }~
e\sinψ=-\dot r_0\sqrt{\frac{R}{GM_E}}
$$
This is sufficient to calculate $e$ and $ψ$ as polar coordinates and thus the minimal and maximal elevations, periapsis and apoapsis (or perigee and apogee)
$$
\frac{R}{1+e}-R_E ~\text{ and }~ \frac{R}{1-e}-R_E
$$
G = 6.67408e-11*1e-9 # km^3 s^-2 kg^-1
M_E = 5.9721986e24 # kg
R_E = 6378.137 # km
H_sat = 500 # km above Earth
V_sat = 36900/3600 # km/s
theta = 0
r0 = R_E + H_sat
dotr0 = V_satm.sin(theta)
dotphi0 = -V_sat/r0m.cos(theta)
R = (r02*dotphi0)2/(GM_E)
wx = R/r0-1; wy = -dotr0(R/(GM_E))0.5
E = (wxwx+wywy)*0.5; psi = m.atan2(wy,wx)
print(f"R={R:.8f} km, e={E:.8f}, psi={psi:.8f} rad")
print(f"H_peri={R/(1+E)-R_E:.8f} km, H_apo={R/(1-E)-R_E:.8f} km")
with the result
R=12469.92166223 km, e=0.81297954, psi=-0.00000000 rad
H_peri=500.00000000 km, H_apo=60298.64124618 km
Setting theta=0.2 gives
R=11977.74005179 km, e=0.82116894, psi=-0.44435938 rad
H_peri=198.81463324 km, H_apo=60599.82661294 km
which is slightly outside the admissible range.
