How the mast step secures to the mast foot depends on the exact design.
I left it vague for that very reason.
The reason for the long tail on the loop is so that you can pull the loop down and detach it, after the mast has been secured.
The checking for the mast being in column and vertical can be done on the water as well as land. Checking for the mast to be in column is relatively easy, as you can sight along a straitedge, like a yardstick, and see if the mast is not in alignment. Checking that it is vertical is usually done by using the main halyard to "measure" the distance to the port and starboard chainplates, and see that they are roughly equal. The reason this is usually more accurate than using a plumb line or level is that most boats don't sit perfect level in the water or on the hard. Rake is usually checked by putting a weight on the main halyard and measuring the distance it is set off from the base of the mast—greater the distance, greater the rake.
BTW, if you get a Loos gauge, get the Pro series gauges, as they're a lot better than the non-pro series and are far easier to use IMHO. Worth every penny IMHO. The way a Loos tension gauge works is it measures the distance it can deflect the wire and translates that into a tension measured in pounds—the higher the tension, the less the wire will deflect.
Finally, get Brian Toss's rigging book, The Complete Rigger's Apprentice. It is an excellent resource, although may take a bit of re-reading to get through some of it.
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You know what the first rule of sailing is? ...Love. You can learn all the math in the 'verse, but you take
a boat to the sea you don't love, she'll shake you off just as sure as the turning of the worlds. Love keeps
her going when she oughta fall down, tells you she's hurting 'fore she keens. Makes her a home.
—Cpt. Mal Reynolds, Serenity (edited)
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Still—DON'T READ THAT POST AGAIN.