BJT Constant Current Driver
How do we determine the internal resistance of the current source?

The internal resistance of the current source is the resistance projected by the collector. Internally, that is defined by this H parameter: \$h_{OE}\$.
Regarding the resistance it projects, because \$h_{OE}\$ is an admittance, the resistance is \$\dfrac{1}{h_{OE}}\$.
This is the dynamic resistance between collector and emitter and it might be in the region of several hundred kΩ.

In which range of UCE is current control possible? When UCE is between
0 and 15V ?

You need at least 2.8 volts from base to 0 volts to forward bias the diodes sufficiently but, diodes are pretty &quot;loose&quot; devices and so you can't expect much in the way of accurate current control at any value of UCE between 3 volts and 15 volts. I mean, it may be OK for what you want but, for someone else with much more stringent requirements, it'll be pretty poor.

Is R1 supposed to be the internal resistance here ?

No.