A question on encoding Assuming there is a machine which can effectively calculate functions not computable by a TM (or the Church-Turing thesis as false) What can we say about aTM solving a problem encoded by this machine...

Q1.  Is there a not trivial example of a Turing decidable problem which still remains decidable once encoded with a non-Turing computable function?

Q2. Is there a counter example?
Q1. The empty language is decidable however you encode it.

Q2. Represent the natural number $n$ as the encoding of the $n$th Turing machine that halts when started with a blank tape. Under this encoding, any set of natural numbers, except for $0$ and $\mathbb{N}$ is undecidable.