First, "
Let (Neil Olshey) worry about how we manage our cap and how guys get paid."
But to answer a few of the questions, Draco's right about a lot of it.
1. You have to be under contract with one team for 3 consecutive years to be considered a "bird exception free agent" (or 2 for "early bird"--which we don't need to go into here). All that means is that a) the team can go over its salary cap amount to re-sign you, and b) you can have a maximum contract of 5 years instead of 4 and at 7.5% raises instead of 4.5%.
2. JJ's case was a bit odd (same with Novak, Lin and Billups) because normally, you have to be with one team for 3 years (or be traded, in which case your Bird Rights follow you). The CBA is pretty explicit that if you CLEAR waivers, your Bird rights are gone. But those guys all were picked up on waivers before clearing, so they kind of fell into a loophole, and the arbitrator ruled that they could retain whatever Bird rights they'd earned (full Bird for JJ, early Bird for Lin).
3. But as Draco said, we didn't "use" Bird to re-sign JJ. He was renounced to give us more space (removing his free-agent cap hold of 4.3M or so) and became a UFA. If we would've gone over the cap signing anyone, then we couldn't have offered JJ any more than the Room Exception. But we had space, and he signed with us on a 1/4M deal. Technically, right this second, he doesn't have any Bird rights, which is why he had a no-trade clause. He will at the end of the year, which, of course, is when you need them.
Non-Bird FA are more restricted in the contracts they can get. See Coon #25 for more writeup on it.
4. The unrenounce-the-renounce aspect. If we would've renounced JJ to get enough space to sign Hibbert, and Indiana matched, then we could have rescinded the renouncement and had his free-agent cap hold of 4.33M go back on our books, reducing the amount by which we could sign UFAs. But we didn't do that. For instance, this summer we can have a 5.3M cap hold for Maynor and 7.6M for JJ (190% of his previous contract). This effectively means that we have no cap space. But let's say we negotiate with RFA Tyreke Evans for a deal starting at 7.5M. We can renounce JJ to offer Tyreke the deal, which he signs. SAC says they match, so we can then rescind JJ's renouncement and have his FA cap hold put back on our books.