I set up a test with Agenda as follows
var agenda = new Agenda();
agenda.database(config.db.host + ':' + config.db.port + '/' + config.db.name, 'jobs')
var i = 0;
agenda.define('some job', function(job) {
i++;
console.log( i+ " Run at " + new Date().getMinutes() + ":" + new Date().getSeconds());
});
agenda.every('10 seconds', 'some job');
agenda.start();
Output is as follows -
1 Run at 56:59
2 Run at 57:13
3 Run at 57:28
4 Run at 57:43
5 Run at 57:58
6 Run at 58:13
and so on...
I left the job running for 3 hrs and I still see it running. I haven't seen any of your first three observations. I do see a delay of ~5s for each run as you can see by the second print. This could be because of mongo latency due to the remote database I am hitting and / or due to log printing delay. I have used Agenda before and I don't think you should use it if you need to execute your job at a precision of seconds.
As an aside, why would you use a scheduler for a now event, won't you just use console.log() when you need to log? If you need to do something else async, isn't it a simple call.