Question 1: During the escape Cat lures the 'bots away from the fleeing Sally and Antoine. Why didn't Sally notice that Cat was gone, and when her and Ant stopped at the at the regroup area they seem to not care about their missing comrade. Not only that, but when Sonic brings it to their attention they still don't seem to show any concern. Why is this? Which leads into question 2:
Likely this isn't the first time Freedom Fighters got separated while escaping. Also they probably did notice but couldn't go back for him.
Question 2: When Sonic proclaims that he is intending on rescuing Cat both Sally and Antoine still aren't worried for his welfare and not only that Sally says that she intends on going into the swamp to look for her father instead. Does this show her as a bad leader with tunnel vision? Wouldn't it make more sense to first save Cat and then go looking for The King with a proper expedition? I mean, he's been wherever he is for ten years and waiting a day or so wouldn't change anything.
If your father that was missing for ten years who you thought was dead might be at a certain place, would you logically wanna wait? Sure it isn't the most logical choice, but it isn't a case about logic here. Sally isn't perfect either, she can make the wrong decision.
Also combining the two questions, the King was likely on Sally's mind so she wasn't paying attention to Cat when he disappeared. Antoine....well likely he was checking Sally's butt out. 
Question #3: After being taken before Dr. Robotnik Cat is threatened by Robotnik saying that he has "amazing machines" that can "do" things... What was he referring to was he intending on taking the information via some sort of robotization like process or is does he mean old fashioned torture?
Threats. He is threatening Torture to test Cat's willpower. Likely he wants Cat to sweat it out as kind of a mental torture.
Question #4: When Sonic finds Cat in the cell Cat informs Sonic that Robotnik's surveillance has picked up Sally. How did Cat find this information? Is it standard procedure to inform captives of up-to-date intelligence?
My guess is a Priority One Freedom Fighter interrupted Cat and Robotnik's "chat".
Question #5: Cat tells Sonic to not worry about him and help Sally. Why didn't Sonic at least break open Cat's cell? In the previous scene we saw Sonic spin-dashing through doors; it wouldn't take but a second to give Cat a fighting chance to get out.
You forget Cat is old. Likely Sonic could escape but Cat would have gotten gunned down.
Question #6: Why did Sally wait until she had been slogging along for what appears to be a great deal of time before asking NICHOLE about any structures in the swamp and only ask after Antoine of all people asks where they are going?
....yeah that one was just.....
....
What the heck Sally?! You even knew what the place was.
Question #7: Now, I could go on and on about how Sonic Boom and Blast to the Past contradict each other. Just to make it simple, how would you explain away the fact that The King left this trail of messages for Sally and parts of his cape behind when Robotnik overthrew him in a surprise blitzkrieg and made The King do "The Time Warp" in The Void? (I couldn't pass up the Rocky Horror reference seeing as Tim Curry being in both casts!) Again, I have my own personal theory that I will wait to tell.
Likely when Sonic and Sally went to the past history changed.
Also possible is that the message was recorded ahead of time and Sally only found it long after it was intended to be found.
Question #8: Was Ironlock really built in 1543 AD? Or was it by another calendar? Also it was abandoned 100 years ago, does this mean that Mobians have been using a structure that exists in our time?! And who was it that was being held in that prison a century prior?
I got nuthin. I don't really consider Mobius Future Earth so....
Question #9: Why was there a booby trap in front of the computer that drops people into the sewer? Wasn't this a prison cell meant to keep prisoners locked up?
Likely to avoid having escaping prisoners access to the computers.
Question #10: I believe that it has been covered before in this thread, but who built the robotic alligator and what's its purpose?
Moat Monster. Guard Dog. Likely designed to scare people from escaping.
Question #11: What became of our honorary Redshirt Cat? Killed, Robot, or something else?
That's what I'd like to know. I'm a sucker for expanding Minor characters' backstories.
Question #12: Where did Sonic procure that spy orb?
I don't know when he picked it up, but likely he snagged one after he found out Cat was gone.
Support for this is he tells Robotnik the StealthBots' destruction was for Cat.
Question #13: Finally, since when is sabotaging four stealthbots such a big victory and recompense for a lost comrade? (Cat) What's stopping Robotnik from just building more and doesn't it stand to reason that he has more that just four of them?
Five. Sonic killed one on his own.
Likely because the Freedom Fighters did defeat all 5, Robotnik deemed their design inferior to their intended purpose so decided to not waste time remaking what the Freedom Fighters already could defeat.