The integral can be relatively fast coped with, by applying partial integration:
integral v d (v*gamma ) = v^2 * gamma – integral (v*gamma) dv
and then using a substitution say u =(v/c)^2. One has then to integrate 1/sqrt(1-u) over u, which is simple.

A remark on the fact that the Lorentz transformations can be expressed with the help of (finite) differences [Delta’s] in their formula: this is a consequence of the linearity of the transformation. Also related to: Space-time is in a certain sense flat in Special Relativity.

Ok

The physical meaning of “now” is in my opinion very different from our intuitive ideas. In the first place, we cannot directly observe (or ‘see’) the things (events) happening at other places in the “now”, since the information about them cannot instantaneously come to ourselves. The “now” of an observer is a mathematical construct, a bridge between past and future of the observer.
Also the aliens far away will have a “now” that is not as a whole immediately accessible (seeable) to them. In fact, as I understand the result of special relativity, the theater of all events and processes is a manifold where space and time cannot be separated in two disjunct pieces (in contrast: Newton thought about time as being an absolute time which is not dependent on reference frames). The time flow of some person, – with past, now and future -, is a somewhat arbitrary chosen direction of movement on that 4-dimensional manifold, and when comparing the time flows of different persons, the velocity with respect to each other has to be taken into account. The “past” of one person can possibly be the (unknown) “now” of an other being, and that is also true for “the future”. So in the story of the special relativity, there is on the manifold no absolute distinction (compartimentalization) between past, now and future.

I do not understand the given numbers and numerical results, with such a low velocity as 30m/sec in the example of George and Sarah when measuring the length of the passing train and discussing the length contraction: the result should be derivable by applying the Lorentz contraction formula. Is there a mistake in these numbers (specially, the velocity of the train)?

I think that the reasoning implicitly uses more than two clocks, namely: the moving clock and (at least) two clocks in the rest frame, – at different places (!) -, which are synchronized with each other, registering respectively the departure and the arrival of the photon in the moving clock, on their respective positions in the rest frame.
Why I think this is essential? Because if there are only two clocks then one can reverse the argumentation and consider the moving clock as being at rest in his frame and the other clock as in fact moving, and then one gets lost (in my opinion). But in the reversed story “the additional clock” is in fact an other one.