And, but, so

These words are built on 그, the 'that' from day 4, plus a form of a verb meaning 'to be so' or 'to do so'. 그리고 is roughly 'doing that, and', 그래서 'it being so, therefore'. That is why they come at the start of a sentence: they point back at the whole of the previous one, the way 'that' does in 'That is why I cannot go'. Each sentence before them is complete and keeps its own tense.

그런데 does not really mean 'but'; it means something closer to 'given that'. It lays the first sentence down as background, and what follows can be a contrast, a new topic or a question. That is why one word covers both 'but' and 'by the way'. 하지만 is a pure contrast with no other job, so it is the clearer choice in writing, while in conversation 근데 does almost all the work.

-고 turns the first clause into a link in a chain rather than a finished sentence, so the ending of the last verb, with its tense and politeness, covers everything before it. This is the same design as the rest of Korean: the important information waits until the end. Because 고 attaches to the bare stem, there is no vowel rule and no irregular change, which makes it one of the easiest endings in the language. Use -고 when two things belong together, and a separate word when you want a pause between them.