DD0CW about the observed 24h period: I would guess most of it is doppler. Neither is inclination exactly 0 deg (as of now, it is 0.01 deg), so this results in an apparent North-South movement over 24h. Next is the non-circular orbit, it is slightly elliptic. Perigee is as of today 35789.0 km and Apogee 35799.5 km. So this results in a West-East movement as part of the orbit is slower than perfect and another part is faster than perfect. And on top of this, the different heights result in a up-down movement of course, hi.
You can actually have it calculated.. just load the TLEs in any satellite tracking program. Take into account, that the beacons are generated on the ground so there is actually the sum of uplink doppler (on 2.4 GHz) plus downlink doppler (on 10.5 GHz).
pe1hzg No NDA this time: I don't know.. but for commercial transponders, they can actually do geolocation of any uplinking stations to a scary degree of precision.. maybe it's linked to that.