Does the Greek syllable “καθ…” at the beginning of so many Greek words have any significance or meaning?

As others have mentioned, kath– is a variant of kata as a prepositional prefix to verbs and verbal nouns. The meaning of kata– as a preposition in compounds is captured in Herbert Weir Smyth, A Greek Grammar for Colleges:

Down from above (καταπίπτειν fall down), back (καταλείπειν leave behind), against, adversely (καταγιγνώσκειν condemn, decide against, καταφρονεῖν despise), completely (καταπετροῦν stone to death, κατεσθίειν eat up), often with an intensive force that cannot be translated. An intransitive verb when compounded with κατά may become transitive.

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