Answers to this can be found by examining the website's code on GitHub.
Apogee and Perigee are numbers that are valid at the epoch of the active TLE for the satellite in question. They are derived from the mean motion and eccentricity values and reference plain old 6371.0 as earth radius.
The altitude comes from the propagator and is in reference to the geodetic ellipsoid.
So, to answer directly:
The altitude can be larger than the apogee from two sources: propagation drift as time passes from the TLE epoch (though this is probably unlikely), and the difference in reference values -- notably, apogee and perigee being in reference to a sphere with radius 6371 km, while the altitude is based on height above the geodetic ellipsoid, which varies between 6378 km and 6356 km, depending on latitude.
To get semimajor axis, use the apogee and perigee values with an earth radius of 6371 km.