Unique product group which is not right orderable 
(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.
 A: Such a group has been found by N. Dunfield, see the appendix to
the paper

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*Steffen Kionke, Jean Raimbault, Nathan Dunfield, On geometric aspects of diffuse groups, Documenta Mathematica, Vol. 21 (2016), 873-915, journal, arXiv:1411.6449.

The group is the fundamental group of a compact hyperbolic three--manifold which has injectivity radius large enough so that it is known to have unique products (and a little more) by a result of Delzant--Bowditch, but Nathan checked "by hand" that it is not left-orderable (by the same method as in his Inventiones paper with D. Calegari,

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*Danny Calegari, Nathan M. Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003) 149-207, doi:10.1007/s00222-002-0271-6, arXiv:math/0203192
which you should check out if you want more examples of non-left/right-orderable groups).
A: Every U.P. group is a t.u.p group. See

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*Andrzej Strojnowski, A note on u.p. groups, Communications in Algebra, 8:3, (1980) 231-234. doi:10.1080/00927878008822456
