If I had all As except for one B first semester senior year and drop to all Bs with one or two Cs in second semester will Tulane rescind my admission/merit aid?
Not a chance. At most, you might get a letter reminding you that college work is definitely harder and requires a lot more self discipline than at high school. Even that is unlikely as long as you avoid D’s. But even if they don’t send such a letter, it is entirely correct.
Lots of people go through senioritus. Just make sure there is no hangover, as there are tons of temptations to procrastinate at college. End of lecture. 
@fallenchemist is correct. Tulane wants you, or it would not have accepted you, so it will take a lot to get rescinded. Don’t get D’s, don’t get arrested, and don’t be found to have lied on your application, and you will be fine. Although, I would work to avoid any C’s so you don’t get a warning letter.
If I got a warning letter would I be on their watch list? Would they keep a closer eye on me in college?
This isn’t the TSA; there is no watch list. However, if you got a warning letter (and I would not even worry about it until or if it happens), you should expect to take it seriously.
Once the class is admitted, admissions is on to the next cycle. There is no “permanent record” that will follow you around for life.
@skieurope said: “This isn’t the TSA; there is no watch list”
This made me LOL.
In your case, if things get no worse, I feel confident skieurope is right. Lest other readers overgeneralize, however, there are cases short of recision where students can be placed on academic probation before even showing up on campus. That would indeed be a kind of “watch list”, but that usually requires failing something or getting a couple of D’s. There is no universal policy, each school makes their own judgement, but I have never seen any action in a case such as yours. But for those wondering, a school could require that you get at least a 2.5 GPA your first semester or year to stay in school, where normally it would be a 2.0. Or they might just require you to meet with an advisor every other week or once a week, where normally there is no required schedule. Things like that.
Many schools have various ways of “keeping an eye on you” for any number of reasons. A letter short of probation (which I feel we need to keep repeating seems unlikely for you at this point) would certainly be one and might cause your academic advisor to make a note to check with you earlier than they might otherwise. There is an old principle that applies to schools, wherein they are acting in loco parentis. This is an actual legal issue, but also an ethical one that some schools follow more stringently than others. You can read about it on Wiki. https://en.wikipedia.org/wiki/In_loco_parentis Other, non-academic, examples are that many schools will react to parental requests that a school counselor call regularly if you have a history of depression, or have your RA be alerted if they want to make sure you are taking medications you need. So while the starting point for most colleges is to leave you alone as long as you are acting like an adult (loosely speaking 
), most are prepared to work with you and your family when an assist is the difference between a successful college experience versus something less.