Sorry about some spelling mistakes in my earlier reply.
"the purpose of the quill shaft drive is to absorb the axial movement of both units". I do not think that this is the case. The gearbox will act as a locator (sorry cannot think of a better word) for both high speed (turbine) and low speed (generator) shafts.
High speed. If you look at item 305, someone has written 'tilt pads' on the drawing. these form a thrust bearing,320 and 305 are journal bearings which support the high speed pinion gear 1. The thrust bearing locates the high speed shaft system. It is located at the best point in the shaft system, since these gears are helical, which gives many advantages (google it!) but results in some thrust from the gear, so there needs to be a thrust bearing to locate the gear, as close to it as possible.
The turbine drive shaft is bolted to the flange at the left hand end of 320. So the turbine is also located by the tilting pad thrust bearing and will grow to the left.
Low speed. The generator driven flange is bolted to the drive flange at the right hand side of the low speed shaft (marked 'LS' on the drawing). The low speed shaft system is located at the bearing 315, someone has circled what I assume are the thrust faces.
Going back to your questions:
Why a quill shaft? No doubt someone sat down and looked at all the options and decided that this was the most effective way to do this. Please remember that this turbine has a firing temperature of about 800 C and an exhaust temperature of about 500 C. At rest it will be at say 20 C, As the turbine starts and reaches operating temperature, the turbine shaft will move upwards. The shaft system has to remain in alignment and so this is accounted for when the set is installed. The quill will absorb any minor misalignments. From memory, the whole shaft system is a catenary, and the gearbox is 'low' i.e. sits at the bottom of the curve, the generator non drive end bearing is 'high', as is the turbine non drive end bearing.
Spline coupling? I do not think so, looking at 210 and 212, I think that the drawing shows a parallel connection, i.e. 210 is heated up and shrunk fitted onto 212. (Don't try getting it off again!). I think 213 214 are an end plate and locking pin. there may be three pins.
I think that the diagram and table at the right hand side (above 'characteristiques') are the alignment instructions.
That’s about all I can deduce and remember. These shaft systems can be extremely complex system to engineer and can have some difficult operating problems.
As my Technical Director once said: “Anyone with high speed rotating shaft systems will have problems. The object is to get the customer to understand this!”