Why are the words "obviously" and "apparently" often used in repartee? What are some other words that are often used, and for what reasons?
When "obviously" and "apparently" are used to answer a question, the person answering is chiding the person who posed the question. The answerer is saying, as far they are concerned, the answer is obvious and requires little or no explanation or that the answer appears right on the surface and is visible to anyone who views a situation. They are asserting the excellence of their own knowledge and judgment concerning the topic of conversation.
These terms will frequently appear in repartee as the parties to a conversation jockey for rank.
To answer the request for other conversation characterizing words:
"Certainly", "patently" "doubtless" and "without a doubt" are words that a speaker uses to confirm the veracity of their own words. Their use may indicate a desire to assert rank to the other parties of an exchange, but not as strongly as "obviously" or "apparently".
When "seriously" is used, either the speaker is retracting a jocular response that they had made or else they desire to prod the other party to hold up their side better and keep their own jokes in check.
When "truthfully", "honestly", to "tell the truth" are used, you may be a witness to a lie in progress since the default belief in conversation is that all of the parties are speaking truthfully. Those phrases may indicate a party for whom truth-telling is optional, or who wants to convince you of something, the truth be damned. Even when used honestly, there is intention to sell.
When a party says "with all due respect", they are likely on the teetering edge of showing disrespect to the other side of the conversation. If you are only giving what's due, you're not offering your respect without reservation.