If my opponent KO'd one of my Pokemon and then on my next turn I use Origin Forme Dialga VSTAR's "Star Chronos" attack, or if I KO one of my opponent's Pokemon with Medicham V's "Yoga Loop" attack, can I play Raihan on the second of my two turns from either of those attacks?
Yes, you can. Your Pokemon was KO'd "during your opponent's last turn" regardless of how many turns you take since then.
Source: TPCi Rules Team (2022-07-07)
Can I use Raihan if I don't have any basic Energy cards in my discard pile to attach to 1 of my Pokemon?
No, the main effect of Raihan is attaching a basic Energy card from your discard pile to 1 of your Pokemon, so you can't play the card if you can't do that.
Source: Evolving Skies FAQ; TPCi Rules Team (2021-08-26)