Depending on the conditions of your purification buffers, proteins could exist either in different oligomeric structures. They can also form large soluble or insoluble aggregates upon purification. Thus, it is always good to optimize solution conditions for maintaining proteins in their soluble and stable forms.

The dimer and tetramer would elute as 2 peaks if they were not in equilibrium with each other. However, if they are in equilibrium with each other at the concentration present in the column, then the two peaks will converge, making the molecular weight appear to be between the dimer and tetramer molecular weight. If you use a SEC-MALS system, which allows you to determine the mass of the protein at each point during the elution, you would probably see the tetramer mass at the start of the peak and the dimer mass at the end, but a continuum of decreasing masses between those two points. Furthermore, if you change the concentration of protein on the column, you can shift the equilibrium, with a higher concentration increasing the proportion of tetramer and a lower concentration decreasing it.

Very unlikely. What you are observing is the mix population of tetrameric and dimeric species. If the peak is broad, probably, you have more tetrameric population at the begning and more dimeric population at the end. Run the GF with varying protein concentration (Try as much concentrated and as much as diluted). You can also do mild crosslinking with DSS (0.2 mM) and run the GF.