How to treat isomorphic objects in a category Q1. Can a proper class contain an element twice?
Motivation. I wanted to give an example of a groupoid that is not a group, where a group is perversely defined as a groupoid with one object. It seems I merely have to find any groupoid with more than one object. It'd be easy to do so if I just take two of the same object in the category. I suppose this discussion really leads naturally to:
Q2. How does a category distinguish two isomorphic objects?
Q4. Can a category have no objects?
 A: Ok I sort of agree with Lord Shark the Unknown's comment, but I'll try to answer your questions as I understand them.
Q1, This really doesn't make sense. In the sense that yes classes are the same as sets in that regard. But then it seems to me that you're confused about what sets or equality are or something. Here's why I say that: sets (or proper classes) really come down to a single relation $x\in A$. It wouldn't make sense to say for example that "$x$ is in $A$ twice." What would that even mean? Are you saying the statement $x\in A$ two times? Well that's the same as the statement that $x\in A$. Like it's fundamentally meaningless to say something is in $A$ twice.
Now what it seems like you want to do is define a groupoid with two objects, any two objects, say $A$ and $B$, and take a group $G$ in the usual sense, and define $\newcommand{\Hom}{\operatorname{Hom}}\Hom(A,B)=\Hom(B,A)=\varnothing$, and $\Hom(A,A)=\Hom(B,B)=G$. 
Q2, again, I'm not quite sure what you mean. Isomorphic objects are different if they are not equal. Equality is a fundamental semantic notion in mathematics, and just because from the internal language of category theory objects may be indistinguishable, that doesn't mean they aren't actually distinguishable from the point of view of the language of mathematics. Edit: As pointed out in the comments, while this is the case, thinking about equality rather than isomorphism is a terrible thing to do in category theory.
I'll try to give an example. If for example you define $\newcommand{\ZZ}{\mathbb{Z}}\ZZ\times\ZZ$ as the quadruple $(\ZZ\times\ZZ,+,-,(0,0))$, where by $\ZZ\times\ZZ$ I mean the set theoretical product and similarly define $\ZZ\oplus\ZZ$ to be the same thing, then they are equal. However, if you define $\ZZ\times\ZZ$ and $\ZZ\oplus\ZZ$ by their universal properties, then any statement about equality such as $\ZZ\times\ZZ= \ZZ\oplus\ZZ$ doesn't even make sense, because $\ZZ\oplus\ZZ$ is only defined up to unique isomorphism.
Q4, technically I suppose this is more something depends on your definitions, but usually the answer is yes, as indicated in the comments.
A: Let $\mathcal{C}$ be the category of one-point topological spaces.
This is a proper class as 
we can embed $\textrm{Set}$ in it: for a set $x$ define the space $X_x = \{x\}$ with topology $\{X_x, \emptyset\}$, and all of these are different spaces though trivially all morphisms are isomorphisms. 
