I took Kripke to be saying that we can rigidly designate something, and we can rigidly designate anything precisely when we pick that thing out across all possible worlds.
A rigid designator designates the same object in all possible worlds in which that object exists and never designates anything else.
Though an important qualification is that a name that rigidly designates from the standpoint of our world may not be a name that would rigidly designate from a different world, but it does for us.
Proposition 1 seems to follow tautologically from the definition. Proposition 2 seems to suffer from two important ambiguities:
- It is not clear that an event of rigid designation has occurred.
- The meaning of the phrase "over all possible worlds" is unclear vis-a-vis the act of rigid designation.
I could probably write more ... but it's been 8 years since I've read Kripke, and I'm not quite sure where you're going with this.
Or to word it another way in answer to your title question, everything could have a rigid designator if anyone bothered to assign one to it, which often happens in the form of a name that is given to pick it out in all possible worlds from the privileged location of this world.