Embark depth varies greatly, and there are some advice for how to generate thin worlds (but they don't work on pocket worlds, in my experience). It's also common to dig down and miss one or all caverns. The safest was to locate caverns is to dig straight probing staircases that start just below the last know feature (the aquifer if you have one, the first and second cavern in turn after that), as a hole in a cavern roof is much less likely to let dangerous critters into the fortress than sideways ones are.
If your vertical staircase building hits Semi Molten Rock (and thus the digging is cancelled silently) you should inspect the bottom to see if there's hot rock beside the bottom. If there is, build an Up staircase as close to the bottom as possible to block any potential invisible Magma Sea critter upwards diagonal entry into the fortress.
Nowadays I get reasonably thin worlds by setting the advanced world gen parameters to 15 levels above the top cavern (to give me enough space for a fortress below the aquifer [unless it's a conglomerate one reaching down into the first cavern]), two levels between caverns (to allow me to dig tunnels under any cavern lakes without constant cancellation spam), and 3 levels below the 3:rd cavern (to allow me to dig tunnels without cancellation spam from either water above or magma below). Setting these parameters to values other than the default seem to reduce the embark thickness variation considerably and usually be close to what's specified.
I believe it's possible to generate cavern free worlds, but that this requires you to mod dwarves to use surface crops, as no subsurface plants are generated, so vanilla dwarves cannot exist. I've never heard of a case where the number of caverns is different from what the world gen parameters dictate, however.