1) Wine and dine=eating and drinking, usually lavishly
2)Game the system usually means to make it work; in this case however it means how to know the tricks of interviewing and how to get selected.
3)Learning the ropes in boot camps= learn the ways of the work/lessons and the style of them
4) Go to bat= do it, in this case it means to stand in place of you and yes, you are right, to recommend.
Hope this helps!