To directly answer both of your questions:
1) Probably not. Check out this rundown on wall warts. Some have linear regulators and some have switching regulators. In both cases, the transformer and rectifier will output a voltage higher than the desired output, and either a linear regulator or a switching regulator will drop the voltage to the desired one. This is useful because it helps maintain voltage across a range of loads (at least, sort of and in theory).
2) Your range of brightnesses won't vary based on strip voltage because of how the power strip is designed. I go into a lot more detail here about LED strips, but the quick version is this: The LEDs are arranged in bundles. Each bundle has the LEDs in series, allowing them to be powered by 12V (3 LEDs) or 24V (6 LEDs). You put a lot of bundles in parallel and that makes an LED strip. So, you won't get more control over brightness just because the operating voltage is different.
So, you may be asking, "How on earth do I do this then???" Well, it turns out the answer is fairly straightforward. You get a 12V or 24V LED dimmer. These go between your power supply (which provides 12 or 24V) and the LED strip. I've linked one here, to provide an example. I have no experience with that particular unit, and StackExchange isn't for product recommendations anyway, but it should help you find the right unit for your needs.