When Jesus called His disciples and the word says they left "immediately" Matt 4:22, did they actually drop everything "immediately" as we think in the west, or was there some type of cultural interaction between family and the called ones before they actually left to follow Him?
What you intend for 'as we think in the west' is difficult for me to understand. In any case, your question is very interesting and deserves an answer.
The term you focus on is the Greek adverb ευθυς/ευθευς. The basic concept of this term (and the other derivative terms) revolves around the idea of "to go straight" (see please, also the homologous Hebrew term ישרו, used to match to ευθυς/ευθευς, in Mat 3:3, Luk 3:5).
For example, the derivative verb ευθυπλοεω means 'to sail directly', that is, without intermediate stopovers. Similarly, the noun ευθυπλοια means 'a direct shipping, navigation' (see its use made by Strabo, Cercid, Arrian). So, the adverb we analyze (in the context you cited) has the meaning of "plainly", "in a straight out manner", in analogy of a mother saying to her son: 'Go straight home!', her words suggesting his son had to avoid deviations, or intermediate stops along the way home.
The disciples' attitude described in Mat 4:22 is a lesson for us today. We Christians are called to reach their spiritual stature, without put up hurdles between we and the Messiah. Lucian has acutely cited a Jesus' parable (Luk 9:61-62) that proves this concept fully.