No, plants are not unless it says so in their description.
Yes, some creatures are immune to critical hits. Ooozes are immune to critical hits. Creatures with the Aeon subtype, Elemental subtype, Incorporeal creatures are as well, unless the attack is made with a ghost-touch weapon, and swarm subtype creatures. Proteans have a 50% chance to ignore crits. Also note that some of these subtypes also grant an immunity to sneak attacks and/or flanking.
You can also check out this link for more info on plants and other creature types and subtypes.