(1) I am looking for an example of a u.p (unique product) group which is not right orderable (RO).
Almost any group I pick up (obviously torsion-free, as u.p. group cannot have nontrivial torsion elements) turns out to be RO, whether it be easy groups like integers, modulo $n$ etc, or groups on matrices, or free groups, etc.
An example of a torsion-free non-RO group can be found at this question.
A group $G$ is said to be u.p., if given any two finite nonempty subsets $A$ and $B$ of $G$, there exist at least one element $x$ which has a unique representation $x=ab$ where $a \in A$ and $b \in B$.
Similarly, a two unique product group (t.u.p. group) is a group $G$ such that if given any two finite nonempty subsets $A$ and $B$ of $G$ with $|A|+|B| > 2$, there exist at least two elements $x$ and $y$ which have unique representations $x=ab$ and $y=cd$ where $a,c \in A$ and $b,d \in B$.
(2) Is there a u.p. group which is not t.u.p.? (clearly, every t.u.p. group is a u.p. group).
As every RO-group is a t.u.p. group, which in turn is u.p. group, (1) was a natural question that came to my mind.