For all (d)esignations, it's a matter of pressing the key for start point, scroll (horizontally, or
now across Z-levels however you want) and then press again for the end-point. If you just want a single spot, then it's a double-press with the cursor. But you can also make a single-width line across the map (or a 1x1 'cross-section' straight up/down multiple Zs) or an area or even a
volume designation[1].
There's also the possibility of 'painting' the display with the mouse. Having gone into the (d)esignate (d)igging menu, take the mouse cursor and click (single-click) the spot(s) you want to make a mine. Drag the mouse around for a line, or just go Minesweeper-Mad and click bits and pieces all over. For some things, this is a more useful way of controlling things, but the primary reason I
used to use it (keep the mouse button depressed while changing Z-levels to easily designated a vertical column) is now possible with the key-controls, as mentioned.
The big thing to remember is that if it's solid rock[2] then you would (d)esignate a stairwell in it, and if it's an open space you should (c)onstruct and (b)uild one of the stairwells. Obviously, if your one and only pick is now with the dead miner, then you're going to have to stick with constructions.
However: I'm not
entirely sure whether you can build a down-stair over an open space and then, from there, build a matching up-stair on the level below (you can do the opposite, though, build an up-stair and that gives access to be able to build the matching down-stair on the level above, even if you didn't have any), but apart from that minor caveat you just need to make sure that once you've designate a construction (which it allows you to do as long as there is
currently[3] a spot to access it from, and materials available for it via that route. It does not check to see if any dwarves are able to access that spot. You
may be able to designate the up-stair from the tailings left by the dear-departed miner, but with no dwarves on the miner's level (that being the problem you're trying to solve), none of them will be able to do anything about it, and will only busy themselves (or otherwise) with the various jobs and tasks directly accessible from the surface.
Not quite knowing what the layout of your land is, I'm not sure exactly how I'd solve your situation. If nothing else, you might be able to work out an above-ground fort (can you fell enough wood to build walls in a stockade, at least to make it obvious when thieves/attackers might be approaching?) and perhaps make a (again, wooden) depot, and see if you can get enough trade-goods together to get a pick from the next dwarven caravan (in autumn), all the while practising your food-management (field building/planting/harvesting, food processing, hunting of wild animals, husbandry and butchery of tame ones, perhaos trapping and taming wild animals, etc).
Certainly I chose the name 'Starver' due to my initial klutziness with food and supplies[5], when I first started this game, but in one fort I played, that was plagued by map-wide aquifer (making digging out out more than two or three actual bits of stone beyond my abilities at that time!), my attention was moved to working out the farming system. That was a very good way of being forced to learn!
I think I had other things to say, but probably going on to long already. {Also FuzzyZerling has summarised part of what I said. Naturally.}
[1] But be careful with cross-Z designations of (say) channels, as you could end up undermining or taking out a tall void first that a dwarf working on the upper bits will be forced fall into.
[2] Or enough of a bit of a solid rock to 'extend' an existing dig, such as turn a (dug) up-stair into an up/down-stair combo.
[3] Except for multi-tile designations, where it only initially needs access to
one of the tiles. If/when you start building 'gantries'/floating floorways over space, you'll see this at work.
[5] "Urist McHunter cancels Hunt: Hunting small creatures" was an oft-seen message in my early forts, due to hunger affecting the very dwarves
trying to resolve the fort-wide starvation issues...