Mike,
It may help to think in terms of balancing two items. Kai is a word that is used to connect two words, phrases or clauses together, who function at the same level. With this function it is often called a coordinating conjuntion. There are about 8400 of these in the NT. When Kai joins words or phrases together, it is simple enough and similar to English "and".
When Kai joins clauses and sentences, it is more complicated. Kai can function like an English "then" as in Matt 4:19 (Follow me, and I will make you fishers of people). The English "and" can function the same way. But very often in the NT and more so in the LXX, Kai is a "Semitic kai". It reflects a Hebrew conjunction that simply links two clauses or sentences together and the meaning has to be inferred from context. Quite often, this Semitic kai has traditionally been translated by "and", but it does not function like English "and". You can see that easily if you compare the KJV with, say, the NIV.
The "rule" about anarthrous nouns is misleading.
When the Kai does not balance two items, it functions as an intensifier. It puts a spotlight on the following word (skipping any conjunctions). It still carries the basic sense of "addition". There are about 800 of those in the NT. In most of these (about 600) it is commonly translated "also" to indicate the addition aspect. For instance in Matt 6:10 (as in heaven, also on earth). Sometimes, the word is better translated as "even" in English, but that does not alter the basic usage in Greek. When used as an adverb, you will often find another clause connector, since Kai is not in that case functioning to connect clauses. For instance, you will find hOTI KAI in Matt 8:27 or DE KAI in Matt 10:30 or KAI GAR in Matt 15:27. When it functions as an adverb, it is rarely the first word of a clause, since that position is often taken up by a clause connector.
As you look at texts, you will get a good feel for the difference. Let me finish with one example from Matt 26:73:
Ἀληθῶς καὶ σὺ ἐξ αὐτῶν εἶ, καὶ γὰρ ἡ λαλιά σου δῆλόν σε ποιεῖ.
Surely, you, too, are (one) of them. After all, even/also your speech betrays you.
