the sentence "Doctor Fox would have got on the plane with the paper serviette still sprouting from his collar" hasn't got if clause, can it be the imagination of Elizabeth and can we write it like this: If he was left to himself He would have got on the plane with the paper serviette still sprouting from his collar?
This depends on whether Elizabeth is receiving Doctor Fox after he gets off the plane (the plane trip has already happened), or she is sending him off before he gets on the plane (the trip hasn't happened yet).
If the trip hasn't happened yet, then your interpretation is fine.
If the trip has already happened, then "Doctor Fox would have got on the plane with the paper serviette still sprouting from his collar" means something like "He must have done it, and it was a typical thing for him to do".
Does "reference" mean: a letter that is written by someone who knows you, to describe you and say if you are suitable for a job, course, etc.?
Does "went for her first job" mean "she went for the first job interview"?
It looks that way to me. This is either an old or a foreign variety of English; this use is not acceptable in modern American English.
Does "reservation" mean: the job was not kept for her?
No, it doesn't mean that. This is related to the word "reserved" -- the relevant meaning is sense 1 in Merriam-Webster - "restrained in words or actions". Reservation describes a quality of the letter: its praise of her was, in her opinion, too restrained, or not enthusiastic enough.
Does the whole second bold part mean
Elizabeth is remembering this sequence of events:
She applied for her first job.
Doctor Fox wrote her a letter of recommendation, a "reference".
She read the reference, and it described her as having "an intellect of quite a high order".
She believed that this description was understated. She took this so badly that she cried.
She now views the same description as comical or even complimentary.
He would have got on the plane versus He must have got on the plane
If the plane trip has already happened, Elizabeth can think to herself "He must have got on the plane with the serviette still sprouting from his collar." The use of must presents the idea as something weaker than a fact -- instead, it is a conclusion she has drawn from some evidence.
But she might also think to herself "He would have got on the plane with the serviette still sprouting from his collar." This one is a little harder to describe clearly. I will quote some relevant sections from The Cambridge Grammar of the English Language:
Will vs must
in its central-epistemic use [will] can generally be replaced by must with relatively little change of meaning. Must conveys the idea of conclusion, and is often used in explanations: Ed's late - he must have overslept. With central-epistemic will it is more a matter of assumption or expectation
"Central-epistemic" means a sense of will that relates to how the speaker knows something; the examples given in the book of this use are They will have made the decision last week and (after a knock on the door) That will be the plumber. Essentially, these are predictions with a very high level of confidence.
I might rephrase this by saying that if Elizabeth thinks "He must have got on the plane with the serviette still sprouting from his collar", she probably reached that conclusion by seeing the serviette, whereas if she thinks "He would have got on the plane with the serviette still sprouting from his collar", she might still have reached that conclusion by seeing the serviette, but she might have reached it by being familiar with Doctor Fox's personality and assuming that this was the kind of absentminded thing that he would do.
There is another sense of will that might be relevant here:
[40i] He will lie in bed all day, reading trashy novels.
[40ii] Oil will float on water.
Here we are concerned with characteristic or habitual behaviour [...]
Strong stress on the auxiliary conveys the speaker's emotive response to the situation - usually exasperation, disapproval, resignation, or the like: He WILL pour the tea-leaves down the sink.
This is the sense that I had in mind when I said "He would have got on the plane with the serviette still sprouting from his collar" would mean something like "he must have done it, and it was a typical thing for him to do."
I'm sorry if this is hard to read/understand; it's a complex topic and these can be pretty subtle shades of meaning. But I think this kind of thing is interesting.