Point 1. is easy to satisfy even with nontrivial polynomials, since if you know $t$ modulo $2^n$ then you know for sure what the first n transformations will be.

For example. I could take $f=2^{20}x+174762$ and $g=6x+2$. Indeed I choose f such that $3(2f(n)+1)+1)$ is always divisible by $2^{20}$, meaning that the first 21 transformations applied to 2f(n)+1 are
 1. 3x+1 cause 2f(n)+1 is odd
 2. x/2 repeated 20 times cause $3(2f(n)+1)+1)=2^{20}(6n+1)$ is divisible by $2^{20}$.

After this we will have reached k = 6n+1 which is smaller than g(n)=6n+2.

Point two is likely very hard and I have no idea how to tackle that. Indeed answering 1 with polynomials f and g with $\deg g < \deg f$ will already be very difficult.