@jalemkul is correct, GROMACS simply queries the CUDA runtime and driver versions; these are reported as 0.0 or N/A when there are issues in a CUDA setup.
If you want further confirmation please try yourself with your own code calling the cudaDriverGetVersion() and cudaRuntimeGetVersion() functions (or use the test code posted recently here: Gromacs 2020.4 compilation with GPU-support on non-GPU nodes - #5 by jbender).
No that is unlikely. The nvidia-smi output you show earlier has no relevance to the problem, that only tests whether the driver works not whether the CUDA runtime a code is linked against is compatible with the driver API.
Please test your CUDA 11.1 runtime with other software and if that works, we can try to figure out what is wrong with your GROMACS build when using CUDA 11.1.