Do amplified worlds have any flat areas?

Hilltops, plains, valley floors, and plateaus are relatively common also, and make good spots to build. When I did it, I had no trouble finding several nice places within a quick walk.

Yes, there are flat spots. Amplified only appears to significantly alter biomes that are already hilly; swamplands and oceans are two confirmed biomes that are still relatively flat.

I can speak from experience:

_{Before anyone asks, the resource pack is Gerudoku Faithful 32x32.}

You can see the amplified extreme hills are really quite extreme, while the swampland on the right is still flattish. Notice the roofed forest in the middle too: it's also relatively flat, though only relatively as it's still a significant hill in a normal Minecraft world.

Notice that even the swampland is ever so slightly amplified:

You can see that though the swampland is fairly flat, unlike in a normal Minecraft world where it's almost entirely flat, the swampland here has variations in elevation up to 6 or 7 blocks. The amplified math probably has a very slight effect on swampland because the normal swampland biome probably has a very tiny height variable that's only notable when it is extremely amplified like this.

So if you can find a swamp, you'll have plenty of room to build. Whether you can find one in as brief a time as 10 minutes isn't something we can answer; even in a normal Minecraft world, it usually takes longer than that to explore just the biome you spawn in and its neighbours, let alone any farther. Though it's entirely random how long it will take (since it depends on your world's seed, where you spawn, and which direction you walk in), swamps are not especially uncommon and you should be able to find one in a reasonable amount of time, for some definition of "reasonable" – and as a bonus, a house built there will have spectacular scenery.