Need health insurance advice please!

Here's the situation:

My wife and I just had a baby, and we are on Cobra with Humana. I am in the process of starting up a business and have been on the Cobra since January through my previous job. I believe I've got another 13-14 months for that. We want to eventually get our own family policy, but for some reason have a hard time trusting the folks on the phone. So, here are my questions:

Is there an ideal amount of time to wait after the baby's birth before applying, or does it not matter? I figured it might help to wait to establish that there are no health issues with him.

I'm currently not working, but I've got money. Is this a factor or not?

They made the application process seem difficult. How likely am I to get turned down? If so, is this like a credit record issue that goes on a permanent record? Can I just go around applying to everyone, or will that look bad?

With the baby, our premiums just went up to over $900/month. I don't mind staying with Humana, but am I likely to get a significant break on this? What are typical family policy rates?

Other than possibly lowering our premiums, what other advantages or disadvantages are there to going off Cobra early?

Anything else?

I would really appreciate if anyone could help me with this. Maybe I'm being paranoid, but I just don't want to screw this up. Thank you!