Interesting -- I had never thought about the many ways we use this. In each case I think that even though it's slang, and even though you're right to isolate them as a little different from each other, the slang is a metaphorical meaning that follows logically from the usual meaning of "get" as obtaining or having something.
In the first quotation, you have it correct: "I got this" means "I have the situation controlled [or more commonly, we would say "under control"]." You should NOT think of this as being short for "I [have] got this [under control.]" Rather, you should think of it as a metaphor: i OWN this (situation.) It is in my hands. I can take care of it. Relax. I got this.
In the second quotation, "get" literally means: catch (in the sense of catching a criminal, not in the sense of catching a ball, although those two senses are also related by metaphor). There is a (presumably, I haven't seen the movie) badguy. The goodguys are trying to "get" him and obtain him, bring him under control.
In the third quotation, it's tough to tell without context, but I think that Lucille is trying to answer a question. When she says "I'll get it" she means, "let me just think a little longer, and then I'll have the answer." (So again, "get" means "have." Then when she says "it's gotta be Marv," she means "I am confident that it is Marv based on other things that I know.")
Finally, in the fourth quotation, it sounds like "get" means "defeat." This is not so different from the second meaning; e.g., if you are playing hide and seek, you might say "I'm gonna get you!" instead of "I'm gonna catch you!"