I think Shaw hasn't had many "Real Battles" since he's been a by the book type of StarFleet Captain.
He doesn't have a combat record like Picard or Riker who are famous for winning battles, especially when the odds are stacked against him.
Probably not, if what we've seen so far is any indication. It still doesn't say much for his character if his initial objections about the safety of his crew and ship can be so easily overridden by the suggestion that he might pick up some secondhand fame.