I have a specific question in mind and will provide it if needed but I'm asking in general for now. In any case, a downvote should be accompanied by a suitable comment so should I just comment and not downvote? It does feel odd to downvote a "competing" answer.
The main rule of thumb I would use is to ignore the fact that you answered it, and use regular downvoting rules depending on how you downvote. In one sentence:
If you hadn't answered it, would you downvote?
Especially, you shouldn't downvote early to get your answer up for the OP and others to vote on more, and you shouldn't do this just out of jealousy for another answer. In my opinion, these are strict no-no's.
This has been made community-wiki, so feel free to contribute with edits.
This is really personal preference, but if you did downvote something I would definitely leave a comment saying why you downvoted it.
As for downvoting a competing answer, don't look at it as a competing answer. Look at "Answers" as possible solutions to the problem. The "answer" is chosen by the person who asked the original question, and if they choose the other "answer" then it will display at the top regardless of your downvote.
You shouldn't downvote to be vindictive, but you can absolutely downvote answers that don't answer the question or are just bad answers. Downvotes cost you rep, which encourages judicious use of them, but if you think the downvote is warranted the fact that you have a better answer shouldn't stop you.
If you think, the proposed answer is not really an answer, i.e It doesn't do anything with the question asked (And you are sure about this), Then it is OK to downvote it. My personal opinion is, It is your duty to downvote a really bad answer. But... If you have hesitation about the answer's usefulness, Don't downvote that answer. It is good to see all answers in the same view, regardless of whether they are competing or not.
First point I'd like to make. This is not a competition. There is no competition here. We are all about helping each other.
As for down voting answers to the same question to which you have answered, for me is not a necessity. How would you feel if your answer will be down voted?
I would agree to down voting answers as well as questions if and only if the answer is NOT useful, and if it would also result to more harm than good to the asker. Other than that, you should just leave a comment.