I think just illustrating some of your examples and the ones of Abbott-Smith's adduced in the comment should illustrate the adjunctive καί, which has survived into modern Greek after 2 millennia. I feel bad I cannot find the analog English usage including and.
A rough-and-ready rule is to translate the adjunctive as "as well", or equivalent, or "even"; a non-primary tack-on to a list, so with subordinate or hypothetical status. English doesn't use "and" for that, but, as I indicated, some other languages do...
ἐγὼ δὲ λέγω ὑμῖν μὴ ἀντιστῆναι τῷ πονηρῷ: ἀλλ' ὅστις σε ῥαπίζει εἰς τὴν δεξιὰν σιαγόνα [σου], στρέψον αὐτῷ καὶ τὴν ἄλλην: [even the other cheek, as well]
Mt:40. καὶ τῷ θέλοντί σοι κριθῆναι καὶ τὸν χιτῶνά σου λαβεῖν, ἄφες αὐτῷ καὶ τὸ ἱμάτιον: [hand over even your coat, as well]
Mt 20:4. καὶ ἐκείνοις εἶπεν, Ὑπάγετε καὶ ὑμεῖς εἰς τὸν ἀμπελῶνα, καὶ ὃ ἐὰν ᾖ δίκαιον δώσω ὑμῖν. [you, as well, go to my vineyard]
Mk 2:20. ὥστε κύριός ἐστιν ὁ υἱὸς τοῦ ἀνθρώπου καὶ τοῦ σαββάτου. [Lord even of the Sabbath]
J 7:47. ἀπεκρίθησαν οὖν αὐτοῖς οἱ Φαρισαῖοι, Μὴ καὶ ὑμεῖς πεπλάνησθε; [have you been deceived as well?]
J 14:7. εἰ ἐγνώκατέ με, καὶ τὸν πατέρα μου γνώσεσθε: καὶ ἀπ' ἄρτι γινώσκετε αὐτὸν καὶ ἑωράκατε αὐτόν. [you will know my father as well]
Ro 1:13. οὐ θέλω δὲ ὑμᾶς ἀγνοεῖν, ἀδελφοί, ὅτι πολλάκις προεθέμην ἐλθεῖν πρὸς ὑμᾶς, καὶ ἐκωλύθην ἄχρι τοῦ δεῦρο, ἵνα τινὰ καρπὸν σχῶ καὶ ἐν ὑμῖν καθὼς καὶ ἐν τοῖς λοιποῖς ἔθνεσιν. [among you as well. the first καὶ is in between conjunctive and adjunctive, to my ear: as I have been prevented until now]
Personally, I cannot contradistinguish adverbial from adjunctive καὶ .