As I understand, you can use Textsecure as your default SMS app which means it will automatically send either a plain or encrypted message to a recipient depending on whether they have the app installed or not.
Of course if you use a non-stock messaging app, like Handcent for instance, you may not want to switch to Textsecure for other features/reasons.
No, you need to initiate a secure session manually, and my experience was that unless you knew the other person was for sure using text secure, this could cause problems, since of you tried to handshake with the other person using a stock app, it could get stuck in the handshake. This was a while ago, so I don't know if the problem still exists.
Text secure is great, though. Really, really great.
Unfortunately, there are no explanations on webpage and I'm not currently into the mood of reading source code to figure things out.
Maybe outgoing (unencrypted) SMS sent through TextSecure app are tagged as "hey, we can upgrade this to secure comms". Then all you have to do is use the app. But I really don't know.
The current SMS-based version of TextSecure uses whitespace tagging for contact discovery.
The next-gen data-based TextSecure protocol (which is currently deployed in CyanogenMod's WhisperPush implementation of TextSecure, but not yet in the Play Store app) uploads a hashed list of the user's contacts to the server for contact discovery.
Of course if you use a non-stock messaging app, like Handcent for instance, you may not want to switch to Textsecure for other features/reasons.