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.

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If you want $f$ irreducible, take $d=3$ and $f$ with real
irrational root, then use Voloch's answer [here](https://mathoverflow.net/questions/165919/diophantine-equations-with-infinitely-many-large-solutions)

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To the edited question about uniform bound.

I think again no.

In Siksek's answer replace $x-2y$ by $x-2^{2^k}y$. You have the solution $(2^{2^k},1)$ and all the other solutions.