You can open Gtalk from Gmail not the opposite. It is because Gtalk is a part of Gmail. It is not the other way around.
Sign out in gtalk and open gmail and become invisible.
In order to sign out of Gtalk labs, you have to log out of Gmail. The Gmail account is at top of the Gtalk. Signing out Gmail signs out of Gtalk too.
The password at Gtalk is same of Gmail. It is because Gtalk is a subset of Gmail. Gmail is the one that's password is unique.
Yes.Chatting on gmail or gtalk is free.
Gtalk settings are inside the Gmail settings. Gtalk is a part of Gmail so their settings are together. You can find it's settings when you go to account settings.
You can add mobile number in Gtalk easily. Adding one to the Gmail account adds it to the Gtalk. Gtalk contains phone numbers as users can call you.
You have to install the GTalk if it is not available. Gtalk is the instant messaging service of Gmail. It allows sending and receiving messages from Google Contacts.
Gmail and Facebook can be linked. GTalk or Google Talk and Facebook cannot be linked together. It is because Gtalk and Facebook both have their own chatting systems.
Signing up for Gtalk or Google Talk takes the same Gmail ID. One Gmail ID can be used for every Google Product. Be it YouTube, Google Drive, Docs etc.
The time one becomes a member of Gmail. He automatically becomes a member of GTalk. However Gtalk is now replaced by Hangouts.
You cannot record Gtalk video chat automatically. It is not a feature introduced in Gmail. However third party software can do that.
You need to have a gtalk and click they're name then chat or IM