The usual hermeneutic rule is that words are interpreted to mean what they usually mean in the wider culture unless there's good reason to read it differently. For instance, in the New Testament, the word εὐαγγέλιον (gospel) begins to take on a specific sense related to the good news of Jesus Christ. It becomes a technical term.
In order for the term "world" to be limited to a small portion of the world, either the term must be used that way in general or there must be a specific context that requires that limited meaning.
Your first example1 roughly parallels Matthew 28:
18 And Jesus came up and spoke to them, saying, “All authority has been given to Me in heaven and on earth. 19 “Go therefore and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit, 20 teaching them to observe all that I commanded you; and lo, I am with you always, even to the end of the age.”
In this case, the clear view of the text is that the intention is to preach the gospel to all the nations of the world. Looking at how the early church followed that command in Acts, they set out to reach people of all nations wherever they are found. It may take a while ("even to the end of the age") so it doesn't seem out of the question that there may be entire continents that were unknown at the time to the listeners of the command that would still fall under the command.
Genesis is a bit more tricky. I've read about (but not actually read) John Sailhammer's theory that suggests the creation account is mostly an account of the creation of the garden of Eden and not the entire world. (God did create the entire world in Genesis 1:1, but according to this theory there is a gap before Genesis 1:2 which concerns itself with the creation of the Promised Land.) It's possible, under this theory, that the flood would be limited to the portion of the world that was known to be populated at the time. Unfortunately, there's a lot of heavy baggage that comes along with the theory which an interpreter may or may not be willing to carry.
A possible interpretation of the Flood passage is that it represents an observation rather than a declaration of truth. Perhaps Noah and his family got out of the Ark, saw total destruction, and naturally assumed that the entire world was destroyed. When I say that everyone ran for cover when the earth quaked I'm using a similar manner of speaking. There are certainly people who don't live near enough to the epicenter to even feel the earthquake. In the same way, it's possible the entire world wasn't flooded—only the parts the author knew about.
On the whole, I don't see much reason in these two passages to indicate that "world" ought to be used in a local or limited sense. I'd say the burden of proof would be on the interpreter who tries to use a particular word in a non-standard sense.
- The text starting with Mark 16:9 was certainly a late addition to the book. It's quite unlikely that the original text of Mark included the verse you quoted. Just so you know.