I think you need $d>2$.
The answer to the first question is "no". For $F_n$ the Fibonacci numbers, $x,y=F_{2n},F_{2n+1}$ satisfy $x^2+xy-y^2+1=0$
Take the degree $3$ Thue equation $f(x,y)=x(x^2+xy-y^2)$.
For Fibonacci numbers as above take we have $f(x,y)=x$.
Take $h=x$. We have $x \gg x^{1/3}$ and all coefficients are units.
If you want $f$ irreducible, take $d=3$ and $f$ with real irrational root, then use Voloch's answer here