Relationship among Niosii, Avalon MM interface, DMA etc..

Probably the easiest way to think about it is in terms of performance vs. resource utilization. Any of them will work.

Software is the slowest but uses the least resources, assuming you had the processor available already.

DMA is faster than a processor and software, uses more resources, and you still usually need the processor to control it. The DMA can be shared across multiple components.

An embedded Avalon-MM Master which performs it's own (bursting) transfers will be as fast as the DMA (or faster), but the FPGA resources are tied up in your component, and your component is a fair bit more complicated.