To me the placement of けど affects which clause you are contradicting. Both are correct insofar as both are common constructions in colloquial Japanese. If formal writing, you wouldn't want to end a sentence with けど, but you'll find that even in more polite speech it's OK to add some kind of exception or contradiction to a previous statement. Just be sure to up it to like なんですけど or variants.
As for the actual difference in meaning in your examples, to my ears the placement determines which part you are contradicting. For example, if you say "できるけど、めんどくさいよ" then I would interpret that to be something like "I can do it, but it's boring/troublesome/whatever," with the nuance being that you are saying you are able, but that you would very much prefer not to. "できる、めんどくさいけど" would be the other way. It's saying "I can do it, though it is troublesome/boring/whatever," with the nuance being the implication that you may not really want to do it, but you are willing to do it. Think of it as though you were putting the けど first in both sentences, so the second would be "めんどくさいけど、できる。" The difference is a little more obvious that way: you're saying "I can" without wiggling around it like you would be if you said "できるけど." This is what I mean by the location determining which one you are contradicting.
So to get to the point of your question, if you want to make it clear that you can't be bothered to do something, or basically say that you will not do it, your example 1 is better.
I don't see much in your question about でも, but I'll just note that でも can't be used in exactly the same way. It's a little closer in meaning to the English "but," offering a stronger contrast or contradiction, and I would say it comes at the end of a sentence much less often.
I think saying "できるけど" is fine, but instead of めんどくさい you could say like "わざわざしたくない"
You could look here for some other examples of alternate phrasing: http://eow.alc.co.jp/search?q=can%27t+be+bothered