Short answer: Yes, epigenetics play a role in determining gene expression, therefore protein expression and function.
Lifestyle factors like diet, smoking, alcohol consumption, and stress can change one's epigenome. A historical example of this might be the Dutch Hunger Winter of 1944-45; there is evidence that after the parents' generation suffered during World War II, the children who were conceived during this time and exposed in utero to famine had different epigenetic marks than their siblings who were conceived not during the famine.
Epigenetic regulation is also essential for cell differentiation in vivo and in vitro. Embryonic stem cells undergo many changes to their epigenome in order to become mature cells. One interesting example for further reading is the transcription factor NeuroD1, which binds to targets in the DNA to cause widespread changes in gene expression that are specific to neurons, and commits cells to a neuronal fate. It doesn't bind to thousands of neuronal genes; instead, it causes a "ripple effect" thanks to epigenetic mechanisms.
Longer answer: Epigenetic marks are added to the genome for many reasons, including the two you just listed, and I think it is helpful to understand what happens at a molecular level in order to understand "why" we have them and to better visualize what the epigenetic marks ("tags") are. This might help you to think of better examples on your own.
DNA spends most of its time wrapped around proteins called histones. A complex of DNA and histones is called a nucleosome. DNA-in-nucleosomes is called chromatin. Chromatin can then either be "loose" (called euchromatin) or "compact" (called heterochromatin). There are more detailed ways to define how accessible chromatin is, but this is the basic idea. Intuitively, loose chromatin is more easily accessible and the genes there can be transcribed actively. Compact chromatin is less active.
The looseness of chromatin is determined by how closely the nucleosomes interact, which is controlled by the histones' molecular properties. Whether or not the histone proteins interact to make the chromatin compact, or repel each other to make the chromatin loose, depends on what modifications they possess. These modifications are added by histone-modifying enzymes.
Histone-modifying enzymes are regulated by transcription factors (i.e. during the process of differentiation) as well as signals generated by extrinsic factors (i.e. during the organism's lifetime). The balance of activating and silencing activity is what adds and removes these marks to give a cell its individual epigenome.
So, in simple terms, epigenetic marks determine how accessible different parts of the DNA are, and a whole lot of factors (like cell identity and lifestyle) can determine where those epigenetic marks are.