Does the genetic material the sperm carries affect its physical properties
I think there is a strong driving force for sperm to be free living haploid versions of human beings.  Since they are a product of meiosis sperm are the product of recombination of both the male chromosome sets. 

While I'm sure that if we look closely enough, the germline cells that produce the sperm are contributing some protein and structure to the sperm, the life cycle of the sperm is mostly the result of the genome that it contains within itself.  Sperm are prone to defects.  On the average in human beings about half of sperm are non-motile: simply unable to form the flagellum or power its motion adequately for motility. That's a pretty strong selective force in the life of our other, 'half genome' selves.  

Sperm selection - the sorting of which sperm from a single male achieves fertilization is pretty rigorous.  It can take 30 minutes to days.  Once in contact with the ovum, the sperm executes a critical set of transformations and biochemical activations to get past the various layers of the cell.  

Sperm are from the man, but they are each living creatures on their own which we hope will deliver themselves up to the ovum to create another human being.  They vary in their genetic quality and effectiveness and this is highly related to their genetic make up.

Because each sperm is the product of two stages of Meiosis, they can be full of genetic defects, most of which would be fatal in a diploid human being.  The functions of the sperm as it goes through its short but competitive life weed out genetic abberations and help assure that the next generation of diploid human beings is as healthy as they can.  

So they are not DNA boxes that swim.  I'd be hard pressed to think of a passive role for a cell in biology - they are all working pretty hard... I'd be interested in a comment with suggestions of something that is not under selective pressure in biology.   

It sort of reminds me of the old argument about passive mating in females. Behavioral biologists used to think that female animals had little or no active role in mating.   That myth was retired about 30 years ago now.  That link points to an introduction to an entire volume reviewing that whole discussion of active female roles in reproduction in case anyone reading is interested. 

BTW I'm not an expert on gametes, but I'd be surprised if the ova also didn't have a complimentary but quite different selective process they had to get through to become a fully presented candidate for fertilization.  

Just to add a note about how the functioning sperm genes affect the diploid organism: This is one of the most important things that sperm does.  The genes that function in the living spermatazoan are the same genes the diploid organism uses, or at least a substantial number of them.  Further, these genes are scattered across all the chromosomes and so major chromosomal defects and important genes are functional if fertilization is completed.  As an example here's a cool blog post about how mosquito sperm have been shown to use odorant receptors to orient themselves in the fertilziation process. Those same receptors and the signalling genes that link that signal to the sperm behavior are probably quite useful to the adult mosquito.  I'd be surprised if human sperm also did not use similar pathways to function. 

Not to beat a dead horse, but the recent nature has a report of small RNAs in mouse sperm as being an epigenetic carrier for stress.