I think you'd first need to figure out if LinkedIn allows you to update a picture on an account through a web service. If they do, then you'd just need to configure a Web Request to talk to the LinkedIn service. Here is a link to their REST API page. Getting Started: REST API | LinkedIn Developer Network
in generally yes.
but it may depend on environment you are on and you haven't provided any details for.
apart from direct connect and custom development, as Brendan suggests, there are available (for some nintex products) dedicated connectors for different services.
few links that might help you