"Pissing it down" is offensive, but only lightly so. It's the sort of thing most people would try to avoid saying in front of little kids and old ladies, but you do hear it very frequently amongst adults. Occasionally you hear "tamer" versions of this like "weeing/peeing it down"!

A lot of terms for rain are somewhat regional, although I'd say things like "chucking/pouring/tipping it down" are very widespread, as are terms like "drizzle", "spitting", "torrential downpour" and obviously "light/heavy rain". "Raining cats and dogs" is rarely used these days, although you might occasionally find it used humorously/ironically, and if you did use it as a foreigner nobody would think you were weird!

There are some things I'd add to your list, such as "Scotch mist" which is the kind of extremely fine rain that's feels kind of like being inside a cloud, but which can soak you through surprisingly quickly.

"Showers" is an important rain term to understand, it refers specifically to sporadic and unpredictable brief periods of rainfall, and is quite a common pattern. This is one of the things that can be annoying with British weather, you can get showery days when you never really know if you're going to get wet at any particular time, so you have to go out equipped for a range of weather conditions!

To expand on the theme of terms like "chucking it down", some other synonyms for throwing can be used to describe heavy rain, in fact "throwing it down" is sometimes used, but also things like "pelting it down".