I've got three questions here.

1) a binding promise ... -- according to a dictionary it's got the following meaing: binding (v) to fasten ..It means that there is more than just a promise, it is a promise that MUST be kept (bound to uphold; obligated, a duty to keep) You must read ALL the definitions in the dictionary. You just read the first one.

So I guess the meaning of the 'binding promise' is a promise that must be kept. yes.

2) Jocko the walrus -- well, I have no clue on this one...

3) Alex's smile drops -- Does this mean, he stops smiling because he could believe that Henry took his jokes so seriously? I think there is an expression for this, I mean in Alex's point of view... in the meanig of 'non-sense'...like the situation here, you say something for fun but the other person takes it so seriously ...do you know what I mean? anyway... "To "drop" a smile means that he no longer has an upturned face Emotion: smile. So his mouth was just straight across.

  "A binding promise"  --       a  promise  the  keeping   whereof  is     troublesome, hence "binding". The pro-     mise binds  you hand  and foot and li-     mits your freedom.
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In the context you described, I think “binding promise ‘was used as a satire. In leagl terms, "binding" means subject to law but only in writting. Verbal agreement can be leaglly binding but it's circumstantial.
I heard of "binding agreement" which is private in nature and can be a legal document once signed and approved by parties involved.
But binding promise is new to me.
If you heard of the saying “Promise is made to be broken”, then you probably know the legal validity of “binding promise”.
Hello, Goodman!    ...Then how would you re-phrase the opening pharase of the dialog?     (not using the term "binding promise")    I'd do it thus:      "If you promised your girl's father to never see his daughter,    would it create trouble for you?"    What's your version?    Anton
The whole dialog sounds to me like that of a sitcom, starting with "binding promise".
Furhter,does it sound like a promise he would keep?
[Alexa]: If I promised a woman's father I would not see her... I would simply shut my eyes, while she serviced my manhood.
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In this movie, there was an actual walrus. I don't remember this particular scene, but the walrus kissed him on at least one occasion, so it seems the walrus just slapped him with one of his flippers. Or whatever you call them.