Language is a device to transmit meaning. But, as human interactions can defy clear-cut characterization, language can be ambiguous, laced with hidden meaning and innuendo. Sometimes the meaning is to be gleaned from the context - which may or may not exist.
Having said this rather general/philosophical bit let us dive into your first sentence:
Ich kann mir 5 Wochen Urlaub nehmen und dann kann ich mit meinen Kindern verreisen.
"können" means, in a strict sense, having the ability. "Und" just combines two sentences. In a strict sense "und" doesn't establish a causal relationship between the two sentences. "Dann" is either temporal ("am 24. ist Weihnachten, dann werde ich nach Hause fahren") or causal ("wenn A, dann B") and here it can be both.
If the former (5 weeks vacation) would indeed be a prerequisite for the latter (with my kids) one could contract it. This won't really establish a causal relationship, but suggest one nevertheless:
Ich kann mir 5 Wochen Urlaub nehmen und dann mit meinen Kindern verreisen.
The way the sentence is given it is a Aufzählung (enumeration) with two items: 1. take vacation, 2. journey with the kids. How about this:
Ich kann die Fenster putzen und dann kann ich den Boden reinigen und dann kann ich auch das Geschirr abwaschen und außerdem kann ich Wäsche waschen.
Would you really think cleaning the windows is a prerequisite of cleaning the floor and both these chores are necessary prior to do the dishes? I'd rather say this is a potential cleaning person itemizing what he is able to do.
So, having said all this, the point is not
Does "und dann" here mean that if I manage to make use of being able feed 5 donkeys, only then I am able to eat 3 apples?
The question is wrong, IMHO, and therefore there is no answer. The question is not: can I justify this or that interpretation, but: which interpretation could be expected by the average reader/listener?
Ich möchte mir diesen Film ansehen.
It is technically possible to interpret that as "I want to see the movie but that doesn't include listening to its sound".
Coming back to your sentence: I would interpret taking the vacation as a prerequisite for the journey with the kids. Not because of the "und dann" or any other grammatical or syntactical device but simply because I know (by and large) how the world works.
A similar point can be made for your second sentence and I refrain from repeating all I said above mutatis mutandis for that.