Every (rational) integer $n=2,3,...$ is an upper bound, since $x < {\sqrt3} < n$
(since $3<2^2=4$).
Similarly every (rational) integer $n=1,0,-1,-2,-3,...$ is a lower bound, since $n\le1<\sqrt 2<x$.
$\sup B$ DNE. Take any upper bound $r$ for $B$.
Then $r$ is a rational number hence $r\not=\sqrt3$.
If $r<\sqrt3$ we get a contradiction since there are rational
numbers $q$ with $r<q<\sqrt3$ showing that $r$ is not an upper bound. If $r>\sqrt3$ we get a contradiction since there are rational
numbers $q$ with $\sqrt3<q<r$ showing that $r$ is an upper bound, but not the least one, i.e. not $\sup B$).
This part of the question was meanwhile edited, and is now answered in 2.
Here are some details showing for example that
if $r>\sqrt3$ then there are rational
numbers $q$ with $\sqrt3<q<r$.
Note that $\displaystyle\lim_{m\to\infty}(r-\frac1m)=r$, hence for all large enough $m$ we have $\displaystyle\sqrt3<r-\frac1m<r$ and we can take $\displaystyle q=r-\frac1m$ for some $m$.
There ought to be a different proof, involving only arithmetic and inequalities between rational or integer numbers, starting with $3<r^2$, and looking for a positive rational $q$ with $3<q^2<r^2$, but this seems to require some extra work to come up and write down the details.
And ... just in case, $\sqrt2$ and $\sqrt3$ are not rational numbers, that is why they cannot be taken as $\inf$ and $\sup$. They are a lower and an upper bound, and they are $\inf$ and $\sup$ if we are allowed to work with the reals, but if we only consider rational numbers, $\sqrt2$ and $\sqrt3$ do not belong and cannot be used. There are well-known proofs (since antiquity) that $\sqrt2$ is not a rational number (and there ought to be some of these on MSE).
The relation $<$ is binary, whether you consider it on the reals or on the rational numbers. First it is defined on natural numbers, it could then be extended to positive rational numbers, and to all rational numbers, and to all real numbers. For the positive rational numbers, if $a,b,c,d$ are positive integers, define $\frac ab < \frac cd$ if
$a\cdot d<c\cdot b$. (The extension, from here to all rational numbers is easy, e.g. use things like each negative rational is smaller than each positive rational.)
Once $<$ is defined for all rational numbers, one can talk about lower and upped bounds. There are sets that have a $\sup$ that is a rational number, e.g. $C = \{ x : x\in\mathbb Q,\sqrt2 < x < \sqrt4 \}$, in this case $\sup C =2$. The relation is binary, you compare two numbers and tell which is smaller, e.g. $5<6$ is the same as saying that the pair $(5,6)$ belongs to the $<$ relation.
