Etymology of “that” as both pronoun and conjunction
In Old English, they often used þe as a generic complementizer and relative pronoun. It could stand alone as a relative pronoun of sorts, or it could be added to adverbial phrases to make them conjunctions. For example, for þæm essentially meant therefore (i.e., for this (reason)) and for þæm þe meant because (i.e., for this (reason) that...). More to the point of your question, though, is that it is, of course, just another fossilized case form of þæt or related adverbial formation.
It seems that IE languages originally had three sets of pronouns: interrogative, relative and demonstrative. The neuter stems for each were (with some simplification for the purposes of this answer): *kwo-, *yo- and *to-. In daughter languages, usually one of these ended up dying out and its function was taken over by another one. In Latin and the Romance languages, *kwo- (e.g., quod) was used for both interrogative and relative. In Germanic, *to- (e.g., *þat) was used for demonstrative and relative, while *kwo- (e.g., *hwat) was used only for interrogative. Later, English switched towards using the *hwat based words for relatives as well. This was not the case in Old English, where it patterned like modern German. That is, þæt could be demonstrative or relative, while hwæt was only interrogative. In Old English, you would say se mann þone (þe) ic geseah where as in Modern English, you would say the man who(m) I saw. Note that we still allow the man that I saw, a holdover from an earlier period.
As it happens, ca is no relation to any of the above. It is derived from a Vulgar Latin compound of *ecce hoc, which we could translate as this here. And hoc itself is built from some rare demonstrative stem *gho- plus a common deictic particle *ke. None of those are related to the aforementioned roots.
References:
- A Guide to Old English by Bruce Mitchell, notably sections 168 through 171.
- New Comparative Greek and Latin Grammar by Andrew Sihler, sections 378-384, NB 378a, which discusses the putative origin of relative constructions in PIE.
Don Ringe discusses this stuff too, but I can't find my copy right now.