Yes, when in L1a status you can change jobs. It all depends on finding a company willing to sponsor your visa.
If you change for example to H1b status your sponsoring company normally would request a change of status. After the change of status you cannot work for the L1a company any more. Also, in case you travel outside of the US and apply for a new visa stamp, the consular officer probably invalidates your old visa stamp ("Canceled without prejudice" is what they can put on the visa stamp).
In principle, there is no direct relationship between your current status and the greencard since it is for a job in the future. However, certain visa statuses have advantages like L1a for which you do not have to go through the labor certification process.
You cannot apply for a greencard yourself when in L1a status since it normally is based on a job at your current company. However, if you have recently won a Nobel price or are the a rising star in the NBA there are a few options to apply for your own greencard.
