24V LED Strip Circuit - Switch, Breaker & Wiring Size Requirement
Several of your questions seem to depend on electrical building codes, which are very location-dependent. I won't try to answer those but I wanted to give some general feedback.

You keep saying "technically 13A" and "technically 312 watts" as if you don't believe those numbers and you expect we will tell you to ignore them. Respect those numbers. Even if you don't believe them, design everything as if they were true.

All of the specified limits of your components must be satisfied simultaneously. If the switch is rated at 10A (and in the absence of more detailed specifications) then you must limit your current to less than 10A, regardless of voltage. Running 13A through a switch rated at 10A could have disastrous consequences, and your insurance company may be reluctant to rebuild your house afterward if it finds out what you did.

Likewise, the maximum current rating for a wire does not change because the voltage is low. However, the rating does depend on how the wire is used (inside a wall? in open air?) and on the temperature rating of the wire's insulation. You must consider these factors when you select the wiring. I would never install lighting that passed 13A, much less 21A, through 16 AWG wire. Consult your local electrical codes.

If you are planning to supply power to the strip from one end then you must also make absolutely certain that the strip can safely pass 13A, which would surprise me. If you supply a link to the manufacturer's data sheet you might get better feedback.