Actual Gesture: Pointing to the tip of your nose
It's Function?: It's meant to clarify if the subject of conversation is, indeed, you
Any Associated Phrase?: More like word, that word being 「わたし？(Me?[this also includes any variant of the word you happen to pick up on your travels])」
Anything else?: In contemporary usage, you're more likely to see the version folks in the West use(where someone points to the chest to achieve the same end), and the version first discussed is the more old fashioned iteration; inversely, when the speaker wants to point out who they're speaking with, they point to them, context showing whether or not they mean to be jerks when they do it.
Literal Meaning: "Is your health great?"
Intent: "You alright?(and similar meanings in translation)"
Usage: To ask someone how they're doing, particularly if it doesn't look like they're in good shape, much like the phrase, "Are you OK?"
Notes: It-and other variations of this phrase-can be used to ask both if someone is OK(i.e. If you see someone trip and fall flat on their face in front you, you can ask 「大丈夫ですか？」 to ensure they haven't kicked the bucket), as well as ask permission to do something; it's like when you wanna grab the last donut, but aren't quite sure if everyone else has had one yet, so you ask "Can I grab it?" (いい, in a question, can function the same way)
(Are you doing alright, Chris? Oyama told me you fell off of your bicycle, so-)
(Yeah, I'm doing alright, Ms. Wakayama. I mean my leg kind of aches, but other than that I'll be muy bueno, for sure.)