I believe I'm looking at a conjugation and contraction of the verb かぶる. What is the original conjugation and what is taken out in the contraction?
"かぶせやがって" is just the "-te form" of "かぶせやがる", which you accepted an answer to here so I guess you know the basic meaning.
This use of the "-te" form is essentially an accusation. You often see it combined with "どうするつもりだ" or something like that: "(What do you mean by) + putting a stupid hat like this on me?!". But even when the utterance ends at the "-te" as in your example, the meaning is the same: the dolphin is not happy about having had the hat put on him.
Incidentally, "かぶせる", meaning "put a hat (on someone)", is related to the verb "かぶる", but it is not a "conjugation" of it in the sense that I think you mean. Reason: in modern Japanese you make verbs causative by adding the affix "-(a)seru" to the stem, and that would give you "kabur-(a)seru" in this case, not "kabu-seru". So although "kabuseru" is obviously related to "kaburu", and it even looks like the same "-seru" is used, it's better to consider "kabuseru" as a different word, and not just a "conjugated form" of "kaburu" -- at least in modern Japanese.
(Note that the derived form "kaburaseru" is actually used sometimes. I'm not sure if it has different patterns of use from "kabuseru" or whether it should just be considered a variation that arose precisely because "kabuseru" can't be derived from "kaburu" directly in modern Japanese.)
In your last example, ”かぶせん" is a contracted version of "かぶせるん" (which arguably is itself a contracted version of "かぶせるの"). る and るん turning into ん is pretty common in casual speech. "何言ってるの" -> "何言ってんの", "だから言ってるんじゃん" -> "だから言ってんじゃん".
"-n janee" is a standard form way of creating an insulting negative imperative. Here the dolphin is saying something which we might translate into English as "Don't put stupid hats like this on me[, poop-face]!" -- well, ideally something more idiomatic, but I think you get the idea. See this question for some more on that part.