Answer to: What is this part? Any chance it's not crucial?Yes it is crucial, but it is repairable. It needs to be re-soldered into position on its contacts, but the orientation is also important, it must go back on in the same way it was before it came off. Look at the other similar part (just to the left in the picture) and see if it has any markings on it that orient the part. It may also be possible to see the orientation from looking at the part that fell off, look at the bottom contacts and see if there is any way the solder break(s) look like they fit the contact points on the motherboard contacts. You may be able to tell just by trying to re-fit the part onto it's contacts and see if there's only one way it fits into the broken solder points properly. Good luck, and be careful about static, use anti-static measures whenever you work with electronic parts.
Answer to: When earphones are plugged in the main speakers are not longer mutedThis is probably not the issue, it's usually a software issue. Either the setting for "Jack detection" is turned off in the sound card options, or the sound card drivers may need to be reinstalled (or updated and reinstalled). You can find the sound options in the Control Panel of whatever Operating System you are using.
Answer to: How to cleaning a graphic card?You seem to forget the single most important thing to do while working with these types of electronic devices...STAY GROUNDED!! The very first thing a person MUST do before touching a single thing is to wear an anti-static wrist strap and connect it to a metal part of the machine you're working on. Static discharge is the NUMBER 1 killer of electronic parts! Even after you've removed your graphics card from the machine, STAY GROUNDED, and stay that way throughout the entire process.