I'm talking about the 'fighting' part of the budget more than anything else. How much money and effort around the world is spent on destructive or defensive behavior rather than constructive or exploratory action?
Very little. Most of the military budget goes to research of one kind or another. The rest goes to pay salaries for all the military employees.
Are you just making that up? R&D is around 10% of the US military budget (and the US military budget is roughly 50% of the world budget).
If the construction, deployment, staffing, etc was pointed towards space research and exploration, we'd be able to push forward all sorts of projects.
Well, we've still got LIGO carefully looking for gravity waves. So far we haven't detected any. Maybe it'd be a good idea to make one hundred percent sure that they actually exist (or else that they're definitely below the limits of LIGO) before building another expensive instrument. There's no real hurry. Gravity waves will still be there in twenty years time if we feel like building LISA.
We have significant evidence to believe that gravitational waves do exist. By measuring the energy of Hulse-Taylor Binaries, we can show that the expected energy loss is there. This was the basis for the 1993 Nobel in physics.
I'd say more than significant evidence. They're predicted by General Relativity. If they don't exist, it would mean our understanding of the universe is seriously flawed. Not many theories of gravity can exist without gravity waves. I think the only one people take seriously is string theory. Though, string theory can exist with or without gravitational waves as it's one of the 'tunable' parameters.