Wink there is no maximum set on our end - if you're talking about the number per room, you can set that in the game settings.
0plus1 which native solution are you planning to use? We've had a few developers integrate it into 'native' games, but I don't believe they're on the app store yet. The API won't request login if there is no internet connection because...well...the API won't be able to load. Right now it's on the developer to check if there's an internet connection and choose to load or not load the API based on that.
If you download the API and include it in your game files, there's already some code that makes sure it can connect to our server and fails gracefully if there's not. If you run into any issues with it though, let me know and I'll get things fixed right away.
For the console.log thing, those all run through a Clay.log method that logs to the console if the object exists and doesn't do anything if it doesn't, so that shouldn't be an issue.
I don't get the unsafe access message when I try out the API & login modal aside from Facebook (screenshot). Could you expand the message and let me know which method it's coming from? All the cross domain stuff should work just fine, so it's still usable on a native app.