Yes, for all of the above. In fact all cats can purr, however the big cats in genus Panthera can only do so on exhalation ( apparently due to an anatomical difference in the hyoid apparatus ).
‘True’ purring appears to be limited to the Felidae and Viverridae ( civets, genets, and such ). Their are analogous sounds produced by other mammal groups, but via a different mechanism, so some have argued it should be called something else.
- Tamerlane