Indeed, have you seen that you can activate additional tagging presets as visible in @SekeRob 's answer? You can also put presets that you use often on custom buttons for quick access. Personally, after using the osm tools for a while and being familiar with many tags, I find it easiest to type the tags rather than clicking through presets, thanks to autocompletion this is very fast. The keyboard shortcut to add a tag is ALT + a (or OPTION + a on Mac)
Pasting tags in JOSM is done with CTRL + SHIFT + V (ctrl+v is already pasting whole objects, so it cannot be the same, on mac, it is CMD + SHIFT + V).
If you are on a Mac, shortcuts are kind of a mess, I agree. Basically you have to remember which shortcuts use CMD and which use CTRL. For pasting text into text fields, you must use CTRL + V (not command, also on Mac).