Keep in mind that your estimate is also for purely earthlike worlds capable of sustaining life on their own. Add in some simple terraforming of marslike planets and that number jumps dramatically. Add in spaceborne habitations orbiting stars and the number begins approaching something rediculous.
Hell, i don't remember the exact numbers, but a single Dyson Swarm around a Sun-type star would provide enough energy and habitation to support the earth something like a million times over, or something similarly ludicrous. I can't say i really care enough to go find the exact numbers, although they're probably on Wikipedia.
Eh, give it another 50,000 years and the number would cover every star in the milkyway nearly 3 times over.
I'm not saying that we'll suddenly run out of room, then curl up in a ball and die. But as I said, a planet like earth would be very valuable, and not for it's minerals.
*words*
The main flaw with your argument is this: Growth would not be exponential, or at least not in such a simple way. The limits of lightspeed would slow our possible rate of expansion greatly. we could, at most, colonize a sphere of space growing at the speed of light. We would still run out eventually, but in a much greater span of time than you predict. Not to mention, there is no reason to colonize so quickly. Just because we could reproduce so quickly doesn't mean we will. Migrating due to using up resources makes sense, but once a planet cannot support more population... stop making more. It's a simple solution, that might even come up on earth.(we are estimated to still only be using a fraction of the maximum population earth can support)
If we're working under the theory that FTL travel is impossible than any worries about meeting hostile aliens or then meeting us is virtually non-existent anyway.
There is plenty of reason to colonize quickly. If an area offers opportunity that the current area doesn't some people will always take that. I'm sure 200 years ago the east coast of america could have support lots of people. But tens of thousands of people still moved west.
As for people just choosing to stop breeding one they hit a large population. Sex is a big biological imperative. However, birthcontrol that far in the future is going to be pretty spot on, so no worries. Motherhood another big biological imperative, also the idea of creating something that lives, thinks, and breathes and will carry on your DNA an in a way help immortalize you is another big one. Frankly, I don't see everyone on the planet up and leaving when they can't have kids, but oh yeah, a good number will move onto greener pastures. Figuratively Speaking.