The closest rendition of 就 into English is "and then" or "and straight away"; "then" can denote a temporal or a logical connection.
就 in sentences 1–4 can be read in the temporal sense, in sentences 1 and 5 in the logical sense. Sentence 6 is a little peculiar because the most natural translation would be "The pony could not even afford to take a sip of water before running way," but if you squint a bit, you could say that 就 here simply means one action is immediately followed by the other in time.
Here are several more examples of 就 as a logical connective:
- 要是我有錢就好了 "if only I were rich" (literally: if I have money, then it is good)
- 聽說她幸福快樂，我就安心了 "I am glad she is living happily ever after" (literally: I heard she lives happily, I am therefore relieved)
- 你睡一覺就舒服點了 "you will feel better after a night's sleep" (literally: you have one nap, then it will be a bit more comfortable); a temporal reading is also possible
The comma is optional in most cases; it is used for rhythm purposes when the second clause is too long. There are, however, are a few set phrases that do not use a comma, 撒腿就跑, 倒頭就睡, 見好就收, 一管就死, 一放就亂 being some of them. I am not sure if 束手就擒 falls under this category because 就 here could also mean "to reach (the state of being caught)."
The phrase "and later" would be closer to 然後, which has a stronger temporal sense.