Your understanding is correct
I was just trying to eliminate any potential confusion about aperture size in your original comment. You said "(a) smaller aperture gives wider depth of field, and a larger aperture gives a narrow depth of field." This is completely correct - BUT - some may confuse "smaller aperture" and "smaller f/stop (aka smaller number)".
For those curious...
The f-stop, aka f-number, f-ratio, is the ratio of the aperture opening to the focal length. The "f" in "f-stop" actually refers to focal length. The "stops" that we see and refer to represent one-full EV (exposure value). So, all other things being equal, adjusting from f/2.8 to f/4.0 will result in a one-EV reduction in exposure.
E.X. say you have a 200mm lens with the aperture set of f/4. Plugging in the numbers and you get 200mm / 4, or an aperture opening of 50mm.
This can get confusing, since the larger f-number results in less exposure
. Few people understand what the "F" in f-stop means and even fewer understand that it represents a ratio, though I find it helps some people to think of it more in terms of a fraction. If a f/4 aperture is equal to 1/4; a f/16 aperture is equal to 1/16 - so the the 1/4 aperture is larger. Using the previous 200mm example, if you set the aperture to f/16, your aperture opening drops down to 200mm / 16 = 12.5mm.
For those wondering why aperture opening is important - it's one of only 3 ways to adjust how much light is captured on an image (the other two is shutter speed - aka how long the shutter curtain stays open, and film speed (aka how sensitive the film or imagine sensor is set to). Being able to "open up" a lens down to f/2.8 regardless of focal length allows for very cool things and is important in fast-moving actions, like sports, racing, wildlife, etc.
You are correct about apertures changing with focal length changes. Consumer grade glass will only typically allow the max aperture at the lowest focal length. Professional series glass will often be able to maintain aperture regardless of focal length (like my beefy Nikkor 24-70mm f/2.8). This is referred to as "constant aperture".