To make a clean work use a coaxial cable stripper: you may find one on aliexpress or rfsupplier site in 5 minutes. They are built for a range of diameters: yours should fit well; I will post a picture of mine when back home end of week. The price is indeed less than 10$ and I don’t think you need anything more expensive. Otherwise a cutter with a sharp new blade will do the job, because you do not have to cut sleeve and braid at different lengths here, just cut through the dielectric neatly to the inner conductor.
How to cut to size: cut it slightly abundant and then trim the inner conductor to size after some trial and error gently pushing into the spring contact. You can shape as hinted the tip to 90° or so, to help centering inside the spring contact and to apply force evenly.
Soldering: it is clearly said not to apply excessive heat not to distort/melt the dielectric. Well, the cable may bring away the heat so the soldering iron shall be about 40W (or up to 60W used gently and wisely) for a good result. Apply some flux paste to the metallic cable shield depending on the material: silvered copper or brass does not need it, otherwise it is not doing any harm. I am using colophony in alcohol.
You cannot rework, it usually does not give good results. If the solder joint surface is not even, you may use a fine grinder gently removing the tin alloy in excess.