Is the RudinKeisler order of ultrafilters linear?
Is it a well ordering?

This would be more suitable as a comment, but I do not have enough reputation points. Among the first results that google spits out for Rudin Keisler is the paper Anatoly Gryzlov: On the RudinKeisler order on ultrafilters; http://dx.doi.org/10.1016/S01668641(96)001095 It claims that there are incomparable ultrafilters (with respect to RK order) and provides several further references. 


As Martin Sleziak already pointed out, there are RudinKeisler incomparable ultrafilters on $\omega$ (while Joel is talking about ultrafilters on a measurable cardinal). This is provable in ZFC. Andreas Blass showed that under Martin's Axiom, there are actually $2^{\mathfrak c}$ pairwise RK incomparable ultrafilters, i.e., as many as you could possibly have.
Here $\mathfrak c$ is $2^{\aleph_0}$.
He also showed that the real line can be embedded into the RK order of ultrafilters on $\omega$ (this again assumes Martin's Axiom). 


One often considers the RudinKeisler order in the large cardinal context of a measurable cardinal $\kappa$, where one considers only $\kappa$complete nonprincipal ultrafilters on $\kappa$. So let me provide the answer for this context, which is that it is independent of ZFC whether the RudinKeisler order on such ultrafilters is linearly ordered. Two ultrafilters are RudinKeisler equivalent if and only if they are isomorphic, if and only if they give rise to the same ultrapower embedding of the universe (I believe this was a topic of some of your previous MO questions). One ultrafilter $\mu$ is RKbelow another $\nu$ if and only if the ultrapower $j_\nu$ by $\nu$ can be factored as $j_\nu=h\circ j_\mu$ for some elementary embedding $h:M_\mu\to M_\nu$. It follows that the RudinKeisler minimal ultrafilters are precisely the normal measures. On the one hand, in the canonical inner model $L[\mu]$ having one measurable cardinal $\kappa$, it turns out that every ultrapower embedding by a $\kappa$complete ultrafilter on $\kappa$ is equivalent to a finite iteration of $\mu$. That is, every $\kappa$complete nonprincipal ultrafilter is RudinKeisler equivalent to a finite product $\mu^n$ of $\mu$ with itself. Such measures form an increasing $\omega$sequence, and so in this model, the RudinKeisler order on measures on $\kappa$ is a linear order isomorphic to $\omega$. On the other hand, in contrast, one can perform forcing so as to create many normal measures. In such a model, the RudinKeisler order cannot be linear, since normal measures, being minimal, are incomparable with respect to the RudinKeisler order. Update. Meanwhile, the RudinKeisler order on this collection of ultrafilters is wellfounded, which fulfills part of what you had requested. The reason is that we can associate to each $\kappa$complete ultrafilter $\mu$ on $\kappa$ the least ordinal $\delta$ generating the whole embedding, that is, for which $M_\mu=\{j_\mu(f)(\delta)\mid f:\kappa\to V\}$. It turns out that $\mu\lt_{RK}\nu$ implies $\delta_\mu\lt \delta_\nu$, and so the RudinKeisler order is wellfounded. 

