Phone-CKI is possible 24-38 hours before departure. So far I mostly was able to check-in in the morning of the day before the flight (regardless on the departure time). This is valid for interconti-flights, I don't care that much about seats on intra-european flights.
Obviously, you don't want to call LH every 5 minutes. You can check the seatmaps on orbitz.com or checkmytrip.com. There you can see, when some seats under airport-control (e.g. row 9 in 747-400) are taken or when the premium sign disappears from the blocked seats (e.g. exit row on many aircrafts).
If you can get them both on the scale at once, no problem.
Not sure about this. Last week, a pax at the F-Class-CKI had this situation in STR and had to repack his luggage.