What height should drop handlebars be - relative to my saddle?
There is no single official accepted guidance. Typically the more you ride, the lower you will tolerate your bars, drops or straights. Pro cyclists have them as low as 10-15 cm below saddle, while amateurs tend to have them level or slightly higher than the saddle.

Don't worry that you cannot spend all your time in drops. Even pro cyclists prefer the more upright position with hands on top of the bars, and will typically go to drops only in critical moments (when in the front, in a breakaway etc.)

The thing with the brakes -- well, that's just the way it is with road bikes and drop handlebars. For giving up immediate access to brakes you gain more hand positions on the bars, which helps immensely on long rides. With your hands in the convenient "on the horns" position, typically you will be able to apply enough pressure to the brakes to slow down, though for emergency braking you'll need to get down to the drops.

You can get an adjustable stem to experiment with handlebar height. Just change it for a normal stem once you decide, or other road cyclists will look down on you -- an adjustable stem is pretty much the mark of a "Fred" :-)