-actic or -acic?

As others have mentioned, climax going with climactic is irregular.

Syntax goes with syntactic because syntax, unlike climax is not a complete word in the classical languages: it comes from Latin syntaxis. This has the common Greek noun suffix -sis, which often can be removed and replaced with the suffix -tic to form an adjective as mentioned in tchrist's answer to What is the adjectival form of "nemesis"? Other examples are prophylaxis, prophylactic; -taxis, -tactic (thermotaxis, thermotactic); praxis, practic. (I don't know enough to explain how this alternation originated, but a relevant Linguistics post is What is the approximate time of the loss of the intervocalic /s/ in Greek?)

Borax and thorax don't come from Greek nouns ending in -sis. The s at the end is just the nominative suffix, not part of the noun stem, so boracic and thoracic are regular derivatives.

Climax also doesn't come from a Greek noun ending in -sis. But this word has become somewhat disconnected from its classical source anyway, the Greek word κλῖμαξ "ladder," applied in rhetoric to a series of words or phrases that build on each other. The OED says "N.E.D. (1889) remarks that senses 1b and 3 ‘are due to popular ignorance and misuse of the learned word’." Sense 3 is the now-common

The culmination, peak, or apex of something; the most important or exciting part of a film, contest, etc., usually happening near the end.

The word "climactic" in my experience refers to this meaning, which it seems first grew common among people who were ignorant of classical languages. So perhaps it's not surprising that the people who coined "climactic" didn't follow the rules that usually apply to learned words.

Also, as GEdgar mentioned, there may have been influence from climacteric which is based on climacter, a Greek word based on climax meaning "ladder rung," also metaphorically "critical point in a person's life."