Yes.

The answer is YES as far as the agencies are concerned, but many times the small print of the contests specify that you should only submit photos that have not been published yet, or only photos taken in 2012.  Also, some contests specify that all images submitted (so not just the winners) can/will be published by the organizers in magazines, websites or books, so I wouldn't submit my bestseller(s) there.

No - if you are exclusive to some agency and your exclusivity contract with the agency prohibits RF distributions outside of the agency.  Many contests have rules which state the submissions will be used at will by the contest operators. By submitting to the contest you are giving copyright or at least unlimited rights of photo use.  This would be considered as an RF distribution outside of your exclusivity contract and would breach the contract.  If you are exclusive somewhere - be very careful with contests.

Read the fine print closely.  Some contests demand a transfer of copyright ownership simply because you submitted.

Exactly.  Read the fine print closely.  As long as there are no restrictions about prior publication, they are not taking ownership of the image and you are not exclusive it should be fine.  Most decent contests nowadays are on top of this - they will say that you must be the copyright holder, have model and property releases as appropriate, and request unlimited use of the image in all media but that you retain copyright.  If it doesn't state something like that explicitly or has any other weird restrictions then back off.    I've made more on some local photo contests than I make on the smaller microstock sites in a year, so those things are worth checking out.  Plus you only need to upload a few images, there is no keywording and you can use images that wouldn't pass QC for microstock so it's usually worth the effort.  Good luck!