>How would you turn a modern western republic into a monarchy?
A restoration can occur in a state whose royal tradition is still alive (see: post-Francoist Spain), but I'm clueless as to how to do this in any other case.
>How would you establish a new aristocracy?
The monarch can, of course, bestow noble titles to whoever he deems worthy.
>Should you take a legitimiste po when reinstating monarchies (bourbons to France, Habsburs to Austria etc) or not?
That'd be ideal, but beggars can't be choosers and a dynasty with a weak claim is still better than a republic. Thus, an Orléaniste/Bonapartiste restoration in France, for example, should still be supported.
>How deal with nations that have never been monarchies like USA or many Latin American ones?
Some, such as Brazil or Mexico, already have claimants, while the rest can either invite some European noble from a notable house or have a strongman proclaim himself a monarchy. This may sound strange from the legitimacy angle, but every dynasty has at its roots someone proving his worth through force.
>Should they be annexed by European monarchies? Is this realistic?
For several reasons, this is out of the question. They could, however, be in personal union with their former European colonizers, as Canada already is.