As far as I've seen, -ka- can be used in any tense and basically means "and then", so without anything indicating the past, surely this could be "and then he cooks and (then) eats" or "and then he will cook and (then) eat" ... no?
According to the instructional preceding the lesson, this sentence would have had to be preceded by some other part of the narrative using the -li tense. But I'm guessing they want us to assume it to keep the lessons short. I don't know about it being used with any tense. One of the things I will ask my Tanzanian friends when I get back there.
This sentence should really have some more context. You never use the consecutive verb form on it's own. There should be something preceding it.
the KA tense is best translated into English with a then statement. Otherwise "she cooked and slept" is rather ambiguous in English as it could just mean "today she cooked and slept" without strictly indicating the order of these activities.
From most of the examples given, it almost seems that the course coordinators were under the impression that leaving off the pronoun from the second verb (which English allows but other languages do not) suggests the -KA- tense. That, as you point out, is not correct. I would note that the second verb here is kula (eat), not kulala (sleep), and we were probably supposed to assume that one does not eat until one has cooked, so that a sequence is implied. On the whole, though, the examples for this -KA- module could use improvement (reported many times over).