Okay, I keep getting told I have no idea of the number involved, so I'll use some numbers to help clear things up. The milky way is theorized to have anywhere from 200-400 billion stars.
Of course not all of these stars have planets orbiting, many are far too large to even have the life 'sweet spot' others, like brown dwarves are much too cold. In many cases you'll have a planet in that sweet spot, that just wasn't able to have those first amino acids form. Let's use Neruz's assumption that 1/10,000 stars have a planet that is harboring life, has a gravity field we can live in, and an average temperature we can live in.
I don't agree with that, but I'll use it for the sake of argument. That's Ten Million planets that can support life. I'll use 2*2 to represent us colonizing more and more as population increases , and then assuming every 'rotation' of that is 5000 years, as in it'll take 5000 years for the planet to reach a population level where large groups migrate to other planets.
2 to the 10th power is 1,024 50,000 years.
2 to the 20th power is 1,048,576 100,000 years.
2 to the 30th power is 1,073,741,824 150,000 years. We'd have run out of planets by now, perhaps colonization of other galaxies is in order?
Certainly there are a few liberties taken here, but the point I'm trying to make is it most certainly would not 'probably take us longer than the lifetime of the universe' to branch out across habitable planets in the galaxy.
Disasters another thing to consider. What do you do as a FTL Civilization when your planet runs out of resources, is on a collision course with a huge asteroid in the near future (Comets, I wouldn't be to worried about, nothing too troublesome about a giant ice cube burning up in the atmosphere.) Frankly, I'd move to a new planet. Not as a single concise group of course.
Black Holes aren't a big worry, you'd have plenty of prior warning before a star collapsed into one, and we can reasonably locate them even with our 'primitive' technology today. Sickness is a good question. I guess I'd have you ask yourself where the field of medicine was at 100 years ago, and to imagine where it will be in tens of thousands of years.
War is an excellent question, and I'm even less comfortable guessing how it'll be fought in the future than I am guessing the rest of this. Though needless to say any war brutal produces lots of wayward refugees.