*Potential Spoiler* Xenomorph in covenant DNA?

Hi dude! David said that the pathogen was very mutagenic and the final result (the mutation) was random but he applied genetics to finally create the xenomorph. I don't know why he designed that way but if you see carefully on his cave, you will see paintings from bugs and other creatures. That makes me think that the xenomorph was just a casual discovery? (I hope the next movies reveal more on that) but also in prometheus you can see paintings of a great xenomorph, that is more interesting because it says us that the engineers knew this final creature (at least one of its multiple variations).

The DNA he utilized to make the xenomorph was apparently from Dr Shaw as we can see in one of the convenant scenes.

Your last two questions maybe are related to David. David create this life forms because he was created to be creative (Walter said that David Model was to human) so He is the result to try to almost replicate human mind. David thinks almost like an human and he has all our abilities (including creativity) so for him this thing represents beauty (for example people have different likes about art, movies, etc) and he considered the alien a perfect organism in the same way Ash considered too: superbly structured, cunning quintessentially violent. Also its defense abilities like the acid blood. 

Thank you Ashtekar for your elaborate explanation! Hence the xenomorph appearance is a random event and also, the egg form is a sort of step forward from the spore propagation method derived from Shaw's ovaries?

That could be the "genetic experimentation" of David. After several tries and continue experimentation with this mutagen, he (maybe) created different forms to create several variations of the alien. One is the eggs another the spores and maybe he created more that the movie didn't show us. 

I believe David tried different variations, cross breeding the pathogen with different animals. I think he is or was after something that can comply with him. Crossbreeding the pathogen with something like a beetle, the results gave him something relatively mindless, that wouldn't comply with him. After experimenting he decided that humans gave the best results.