How do I recover my iPhone from the person that found it in lost mode?

Do not remove it; as long as your phone is linked to your Apple ID it is protected by Activation Lock, which blocks anyone from using it.

Here is Apples lost/stolen recommendations→If your iPhone, iPad, or iPod touch is lost or stolen - Apple Support

Also, beware of a common phishing scam: You may get a text or email from “Apple” or “Apple Security” (or sometimes “Police”) saying your phone has been found and to click on a link to see its location. This is NEVER from Apple or from law enforcement; it is from the thief or fence to steal your Apple ID and identity. Delete any such message without responding to it. Apple never gets involved in lost or stolen devices→Avoid phishing emails, fake ‘virus‘ alerts, phony support calls, and other scams

Not if you put it in Lost mode (if you haven’t already) and don’t remove it from your account. Can you track it, or at least see it, even if it is offline, either from another device or https://icloud.com/find?

If they haven’t contacted you there is nothing you can do. Did you tell them your Apple ID & password, or remove the phone from your account?

I don’t know about updating, but it probably will.