Can I enter a manuscript into a contest if a publisher is already considering it? Would the contest folks and/or the publisher dislike me doing this?
There is no publisher I know of who has a rule saying you can't send your manuscript elsewhere - including into competitions - while they're considering it. However, it's quite possible that the competition will have rules about what you can do with your manuscript so you would need to check those. If the rules say you can't have the manuscript under consideration anywhere else, you can then decide whether or not it's worth it to restrict yourself this way. (I've made my feelings on this subject clear elsewhere.)
If the competition rules don't preclude you having the manuscript under consideration at another publisher, the polite thing to do would be to send a letter to the publisher who's considering it saying that you're planning to enter the manuscript into the competition. That way there are no surprises if they decide they want to make you an offer.