I guess, we could infer that the structure of an amino acid has the same functional units as RNA is used to synthesise it. Therefore, from a logical point of view it would make sense that genes are expressed as proteins. However, why is this the case; why are genes expressed as proteins rather than any other type of bio molecules? Also, from what are other bio molecules expressed if not for genes?
Wow - like a biophilosophy question! The answer lies in in building up the largest diversity with minimal subunits. Efficiency if you will. In this case we are talking about amino acids. We only need 20 of them to approach unfathomable numbers of diversity.
The protein acts as a messenger, as an enzyme, a carrier and it's own building unit. If you wanted that kind of range out of any other known biomolecules, you would need a gene to code for each of them, and itself a gene a completely different mechanism to build them which we can't even fathom. It's so much simpler to code for a protein, which can act as an enzyme to build other biomolecules.
However if you are asking about why the central dogma of molecular biology is the way it is...well then, it seems to be axiomatic.
The fact that genes are expressed as proteins and no other type of molecules is because it is one of the oldest remnants from evolutionary history and it was so successful that it has carried on throughout the eons and is now present in every known living organism. It is so old that most biologists would say that it has been present since life first began.
Scientists conclude that it is one of the oldest because the genetic code is universal. That means the gene that codes for a specific protein can be inserted into a completely different organism and that organism will begin synthesizing that protein. There are very few exceptions to this rule and this is how scientists can culture bacteria to produce human enzymes such as insulin.
Genes do not code for any other molecules because there has been no selective pressure for it (if that can actually apply here) and proteins are quite diverse in structure and function for having so relatively few building blocks. There really are no other macromolecules quite like them.
I am unaware of any non-protein molecules being expressed from genes. You might say that RNA is, but you have to skew the definition of "expressed". Ribozymes are a good example to make this case. But there is a reason the creation of RNA is called "transcription", meaning the RNA is not coded from the DNA, rather it is analogous to it, while the creation of proteins is called "translation", meaning that there is a code involved to get the correct result. It is also obviously clear that RNA serves for protein synthesis and is far less dynamic than proteins.