I've personally used math puzzles and riddles to create a lot of engagement. It is also a skill based game with a promise of specific rewards. All your 3 points can be achieved if you can find a great collection of brainy questions and riddles and other similar challenges. (TED Ed Riddles is a good example) Make it a weekly thing for continued engagement.

Another one would be recommendation bounties. Think about a specific kind of game or movie you want and put on a bounty. Here is an example of a similar event: https://steemit.com/steembasicincome/@vimukthi/share-your-underappreciated-science-fiction-hideen-gems-movies-and-win-steem-basic-income-shares

At least half the people engaged with me on that post were engaging with me for the very first time. It only cost me 11 SBI shares at the end.

A secondary method would be design contest for logos or mascot etc.

Best of Luck!