How can a Shinigami drop its eyeballs? So BB was born with the Shinigami eyes because a Shinigami dropped it's eyeballs. But how is this even possible? Did the Shinigami ripped out his eyes and then let them fall down to Earth? I just don't get it. 

Also, if a human touched the eyes of a Shinigami, would he then also get Shinigami eyes?
a Shinigami dropping it's eyeballs is just something Mello thought of 


  It is unknown how he could have possibly been "born" with the eyes, though Mello wonders if a Shinigami dropped its eyeballs to Earth (instead of a Death Note)


Source - Character, 3rd Paragraph, 3rd Sentence

and since Beyond Birthday only appeared in Death Note: Another Note which occurs before the manga (L's first encounter with Naomi Misora during the Los Angeles "BB Serial Murder Case" mentioned in volume 2 of the manga) i can really only answer this from logical evidence, absent of any official statement and assuming that Mello is correct.

I would suspect but rather than a Shinigami taking it's eyes out on it's own (which are used to allow Shinigami to work out how much life they will gain from taking a life), we can make the assumption that a Shinigami died.

How we know of 2 ways a Shinigami can die


they fail to extend their own life by taking names
they save the life of a human who's death date had arrived


When a Shinigami dies they turn to sand/dust however this isn't instantaneous, 


Rem's Death: Death Note - The Last Name
Gelus Death: Death Note Anime


as we can see from the links above, Rem began to crumble first however Gelus's eye disappeared first

we also know that there is a Portal which allows travel between the Shinigami Realm and earth. i theorize 2 ways that the Shinigami Beyond Birthday got the eyes from died


said Shinigami saved a human which would explain why they were at the portal
said Shinigami was sooo lazy that it waited to the last minute to take a name to add more life and couldn't take a life quick enough before it died


now while we are told that Shinigami are lazy i don't think they are that lazy not to write names (though Ryuk does note this to Light) so i'm going to assume 1)

regardless, the Shinigami's eyes could have dropped out of it's head and rolled out of the portal which could explain why the eyeballs were dropped and not the Death Note (as we see in The Last Name Gelus's Death Note fell to earth however in the anime it didn't).

So then how did Beyond Birthday get the eyeballs? well it was either by coincidence or since Beyond Birthday was at Wammy's House said Shinigami saved Beyond Birthday at birth by saving his mother who may have been pregnant and was about to be murdered.


  Also, if a human touched the eyes of a Shinigami, would he then also get Shinigami eyes?


this i can't answer, i would assume the timeframe is very small because the eyeballs would have turned into sand/dust and Beyond Birthday was just very lucky to have gotten them before hand because if it was a living Shinigami's eyes, well the sight of Light running around with a bag of apples trying to get Ryuk top allow him to poke his eyes, or Misa looking all cute and asking Ryuk if she can doesn't quite match the tone of the series