The answer to the first question is "obviously no", and to the second question is "it highly depends on the category".
Why "obviously no" ? Well you can easily create artifical examples that show that it's not always possible. For instance take a monoid $M$ and view it as a monoidal category with objects elements of $M$, no nonidentity arrows and $\otimes$ the multiplication of $M$. The associator and unitors are the identity (they have to be, because there are no other arrows) and clearly all the axioms are satisfied.
Now in that category a monoid is an element $m$ of $M$ with a map $1\to m$ and a map $mm\to m$ satisfying certain equations. But if there is a map $1\to m$, then by definition $1=m$. So if $M$ is nontrivial, there are no other monoids than $1$ (which is easily seen to be a monoid).
You have other more interesting examples : if you take the category of spaces with the cartesian product, it's a very interesting question of when a space can be given a monoid structure, even up to homotopy. Most of the spheres can't, for instance; if you take a category of endofunctors with composition, monoids are monads, and of course all functors aren't monads; etc.
The answer to your second question follows from that : as you see in my artificial example, there are no monoids except for the monoidal unit (which is always a monoid anyway), in the category of spaces say up to homotopy it is a very interesting question and the answer involves operads, in the category of endofunctors of a category, it is also interesting as it brings about monads naturally; in the category of vector spaces that was suggested, it is easily seen that any vector space has at least one monoid structure, so it's not interesting either.
Therefore "which objects have at least one monoid structure ?" is a question that doesn't have a general answer except for "it really really really depends on the category in question".