I remember reviewing one of these flags from my phone this morning (I do that first thing every morning =), but looking at the timeline there were 3 moderators involved here, at different points in the timeline.
Flagged comments looks something like this in the flag review page:
How should I improve my flags?
@me No, it's a C&P error - OP - Other: This has now been resolved. Please remove the thread. deletedecline
How should I improve my flags?
@OP You can fix it by using ctlk-k - Me - Obsolete deletedecline
Some other question link
Thank you, much appreciated! - Someone - Too chatty deletedecline
How should I improve my flags?
@me Fixed - OP - Obsolete deletedecline
Notice there's little to no context, and that the surrounding non-flagged comments don't show up anywhere; we're not taken to the post unless we go and follow the links and keep a tab for the review page and another for the post. ...which, to be honest, is rather annoying to do on a mobile browser.
Comment flags shouldn't require much context - the reason for flagging them is typically plain obvious just by reading the comment itself. I believe that's the reason why the UI is made that way, so that a moderator can quickly read the flagged comment and decide whether to delete it or decline the flag.
If there's a whole comment thread that needs to be removed, ideally custom-flag the question instead; the UI for question/answer flags is slightly different, and does provide more context.
Personally I don't mind each individual comment (up to a certain extent, say 5-6 comments) being flagged either, when each individual comment taken by itself is obviously obsolete or "too chatty" or whatever you flag it for. If there's surrounding context that's not going to show up in the review page, it's likely going to be missed.
One thing to note is that a custom flag asking to "remove the thread" can very easily be interpreted as a custom flag asking to delete a question (and its answers); such flags are systematically declined, usually with a note from the moderator linking to How does deleting work? on MSE. Therefore I would recommend to avoid that wording and specifically mention that the flag is about the comments. The "thread" lingo is just too context-sensitive to be completely unambiguous in the flag review page. I believe this is what happened with the declined custom flag.
TL;DR:
Should I remove my comments first?
Yes. If your comments are obsolete, you have the power to remove them yourself - no moderator needs to get involved here.
Should I flag every comment, or just one?
For 5 comments, I'd say go ahead and flag the comments that you can't remove yourself as obsolete (assuming that's what they are).
Should I use a different flag?
For 5 comments, no. Flagging as obsolete was correct. If you want to flag a bunch of comments under the same post, custom-flag the post itself.
Should I use a better description in the flag?
Be specific about the fact that it's comments you're flagging to remove; custom flags asking to remove a "thread" are ambiguous and could easily be declined, mistaken for a request to remove a post.