Can you regain an "abandoned" green card?

I'm not sure I understand all the facts, but be careful because not all U.S. immigration lawyers understand the rules related to abandonment. You say you left the U.S. in January 2012. that doesn't necessarily mean you "abandoned" your LPR status. There are a couple ways to definitively now you've abandoned LPR status: either voluntarily filling a Form I-407 or having an Immigration Judge determine that your status has been abandoned. In the absence of those two things, you may or may not have abandoned LPR status due to your stay abroad. The key question is whether that stay was "temporary." See the linked article for more details. Finally, please note that if you have definitively abandoned LPR status then you can't re-activate your LPR status--you would need to reapply for LPR status.

At times, yes

If you still qualify for someone to sponsor you, then yes.