I'll try to address some of your questions and misconceptions about cloaking...
>>do all cloaked ships do that? (in reference to "cloaked vessels that don't sustain damage in combat generally re-cloak after combat with the orbiting ships")
Yes, though I wouldn't say "re-cloak" so much as "stay cloaked". Basically, cloaked ships can't attack planets and can't be attacked, but they can attack. And if they sustain damage, they become uncloaked. This is true whether the damage comes from enemy ships or hitting a mine or something else. And, of course, there are ways to decloak them that don't involve damage.
Here's some light reading:
>> Birds have some kind of special "re-cloak intercept"
Yes, in campaign games, and recently added to Standard games, Birds have an ability called "Cloak and intercept". This is NOT the same as the Deth Specula Stealth's "Recloak Intercept" ability.
>> NOTE: Ships capable of cloaking will cloak while on intercept missions automatically.
No, only the Birds can do this, and it's a separate mission "Cloak and intercept" vs. "Intercept"...they can do either. And only in games with the ability active (Standard, Campaign, some custom games).
>> I believe Bird Super Spy mission ... will also auto-cloak as well,
Yes, cloaking is part of this mission, if the ship can cloak. Non-cloaking ships can still perform it.
(has links to the various versions, like Super Spy Deluxe and Super Spy Command)
>> and Privateer Rob Ship missions will also auto-cloak as well,
Nope. Ships using the Rob mission are vulnerable, but often the goal is to drain the target(s) of fuel completely, and out-of-fuel ships can't fight, keeping your robbers safe...from that ship. If you establish a tow-lock in the same turn you drain all the fuel, you also capture it that same turn. That said, usually your robbing ships are cloaked on the turn before, so they can't be the target of intercept missions, which makes it marginally safer as well.