New answers tagged slang
It sounds like another form of はくしょん, onomatopoeia for sneezing. Searching on Google for things like "ぶえっくしょん" "くしゃみ" seems to confirm that this is the case, although I'm not too familiar with the word myself. I found this page, for example, where it's specifically labeled as a sneeze: 大野「ぶえっくしょん！（←くしゃみ）」 If I had to guess, I'd say it sounds like a ...
How about... ~~する(とき)みたいな(しゃべり)かたをする, ~~するような(口のきき)かたをする, or ~~する(とき)みたいに~~する?
こうぜ is just how いこうぜ gets pronounced in lively "tough guy" kind of speech. It is used in group activities such as sports where a team effort is essential. Te-form of a verb + （い）こうぜ！ = "Let's all (verb), guys!"
You guessed correctly. However, we say さよならだ much more often than さようならだ. The latter sounds pretty dull. (Vowel lengths are of utmost importance to us.) With だ, the speaker is declaring a parting. The speaker would be male almost 100% of the time.
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