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"In various ways...things are getting more convenient for me."
(context: woman is looking for a patient who left his room in a all woman's
"W...Well, I think that he's a person who can manage that much."
んな事 言ってられるか 自分が一番かわいいんだ
んな事 - don't understand this.
自分が一番かわいいんだ - "I'm the cutest."
Second one, personally I think your one seems fine, though if it's me, I'd rather put it as "W-...Well, I bet he should be able to manage that hmself" though. Not the best way to put it,though, so maybe you should also look at what others have to say on this.
Third one, "nna" = "anna". So the actual sentence would be "anna koto itterareru ka". That should help, I guess.