There most certainly is a lot of evidence suggesting related sequences of nucleotides that result in different functional proteins. However, this might be hard to easily see unless you look at specific molecular examples. If you look at a large, evolutionarily-related family of proteins, they usually share a similar overall morphology and often the same, or similar, interactions with other proteins, but the fact that members in the same family can interact with different proteins show they are, in fact, different functional proteins.

One family of proteins I'm familiar with are integrins. Integrins are proteins present on most cell membrane, facing away form the cell. Integrins allow the cell to communicate with cells next to it and integrins let the proteins on the outside of the cell (the extracellular matrix [ECM]) communicate with the cell. All integrins are heterodimers made up of an alpha and beta subunit. There are many different kinds of alpha and beta subunits (numbered alpha1, alpha2, alpha3, etc.) and the specific heterodimer determines what proteins in the ECM the cell interacts with. All the alpha subunits are very similar to each other (and probably originated from one kind of alpha subunit long ago) and so it only took a few amino acid changes to result in a different alpha subunit that interacts with a different beta subunit. As an example, the integrin heterodimer alpha5beta1 is known to bind the protein laminin, while the integrin heterodimer alpha6beta1 does not interact with laminin, but interacts with other ECM proteins that alpha5beta1 does not interact with (i.e. fibronectin and vitronectin). On the larger scale, such seemingly small differences can be quite significant -- if a mouse had a mutation in one alpha integrin subunit, it could have a disease/phenotype very different than if it had a mutation in a different alpha integrin subunit.

By the way, if you zoom out of the protein family scale a bit, it's clear that there are some protein families that are more closely related to other protein families (they probably just had a common ancestor at a much earlier point in evolutionary history).

You are absolutely correct, although I make the caveat that the rate of evolution varies between genes, between species, and over time, so the "molecular clock" that you talk about using to estimate the age of species has to be calibrated rigorously using fossils (which we know the age of independently through various geological means).

However, the co-option of genes that serve one function into serving a different function is a hot topic in evolutionary genetics. This is particularly heavily-studied in developmental genes, that is, genes that control the development of separate organs in plants and animals. The genes, for example, that create a flower in a flowering plant exist in non-flowering seed plants (e.g. pine trees), but they obviously do different things. There is an entire field known as evolutionary developmental biology, or "evo-devo", which is focused on this area. It's also an important process in the evolution of viruses (both beneficial and harmful).