Will DNA nanostructures be useful for medical applications?
I think biomedical applications are not an absurd perspective. More specifically, regarding your concerns:

(1) Unless the DNA is first internalized by cells, and then somehow enters the nucleus (both are not very likely, especially considering the small concentrations we're talking about here after dilution in a human body), you would still need the DNA structure to disassemble to become "available", and the strands to be long enough to be incorporated, which is usually not the case. And, of course, you would need some DNA integration mechanism to be at work, such as a recombination with genome sequences that happen to be very similar to the ones used to build the DNA nanostructure (which can be avoided during the design step). In other words, no, it's quite unlikely. 

(2) Immune responses might actually be a bigger issue; in particular, some Toll-like receptors (TLR), that activate the innate immune system, can respond to DNA and RNA fragments. But immune responses may not be so problematic if the DNA nanostructures are used for short-term mission, like 1-shot delivery to a specific organ, in which case they do not need to stay around for very long. Additionally, this can be a useful feature; immune recognition can be used to direct the trafficking of nanoparticles to lymph nodes for example, where the efficieny of their payload (e.g. an immune modulator) could be maximized.