Is it possible to add new badges or introduce points for sharing links to questions? I think it would be good to encourage more social media sharing of the site - it would help to improve the profile of the BiologySE site, help garner new users, and increase the frequency of questions and (multiple good quality) answers. One way to encourage more social media usage would be to offer points and/or badges. As it stands there are no points available, and there is only three badges (depending on the number of unique IP addresses that read the post [25, 300, 1000]).
Increasing the rewards for specific behaviour also encourages gaming the system to get those rewards, one example are the review badges which were in part responsible for a whole army of robo-reviewers. So we need to take possible negative effects into account.
What could happen if the rewards for sharing were too large is that users start spamming high traffic sites like Reddit to get those rewards. This is fine in moderation, but we don't want to get a reputation for spamming and maybe even be blocked from some sites due to that.
I can guarantee you that giving out reputationn for sharing links won't happen. It doesn't fit with the existing SE philosophy, it would reward users for something that happens outside of the site, and it would likely be too tempting for some users to abuse.
Badges would not be out of the question, but I don't see any good criteria for new badges. The sharing badges already exist, and they span all three types (bronze/silver/gold). Of course one could add badges for sharing x links in total, but that would be rather boring and also not much incentive over the existing badges.
Some more creative badges could be to award them for bringing in a user that becomes a very active member on the site. So e.g. if a user registers after they visited the site via such a shared link, the sharing user would get a badge once the new user reaches x rep. Though I suspect that practical and technical issues would make this kind of badge very unlikely to be ever implemented.
I don't think we should add any substantially higher rewards for sharing links, I'd be too worried about possible negative impacts of such a change.