What is the highest angle the moon ever makes above the horizon at the North Pole?
${R_e}/{R}$ is about 0.017 radian or 1&deg;.
Using your trigonometry and these values:


Lunar orbit inclination = 5.15&deg; (varies between 5.0&deg; and 5.3&deg;)
Lunar perigee = 362600 km, apogee = 405400 km
Earth polar radius = 6357 km
Ecliptic obliquity = 23.44&deg;


I get 27.7&deg; at perigee and 27.8&deg; at apogee.
At that altitude, atmospheric refraction is only about 0.03&deg;.
The highest apparent altitude I found with JPL HORIZONS was 27.91&deg; on 1987-09-15 around 17:15 UT.

Due to nodal precession,
the Moon doesn't reach &plusmn;28.6&deg; geocentric declination every year, only near a major lunar standstill every 18 or 19 years, e.g. in 2006 or 2025.
Near a minor lunar standstill, e.g. in 2015, the Moon's declination is limited to &plusmn;18.3&deg;.