T.S.
You might consider re-wording your expectations to reflect behaviors you want them to SHOW rather than AVOID. For young kids (and for all of us) it can be difficult to simply refrain from a behavior, especially if it is one that is already habitual.
Phrasing positively might also help you get a little more specific about what you want from them. "No competition" could mean a million things. Are you saying they can't play ANY game with a winner? No racing? From your description above I might use "Be a good loser" instead along with "wait your turn" and "give a turn to others". You could also add "Be a kind winner" as the companion piece.
Similarly "use kind words" instead of no bad language etc.
You might also want to give them the possibility of earning MORE stickers per day (but needing a correspondingly higher number for the reward as well). The trouble with just ONE sticker for the WHOLE day is that if they mess up in the morning there is no incentive to IMPROVE their behavior during the day because they've already failed. Maybe one sticker for meeting each expectation, but then they need 50 to earn a treat.
Last, I'd suggest setting the reward ahead of time. "A candy bar", rather than "a candy bar or something small" will avoid having to negotiate about what counts as a reward (which could undermine the whole reward idea if they feel like they didn't get what they wanted).
Hope this helps.