I'll try to answer that before covering a few other things.
CONI is the Italian federation, but unlike some other countries, they are part of the government, thus were able to cross the legal T's and dot the legal I's to make sure they had the power to make the DNA tests legally and match them to documents from Operation Puerto legally.
REFC is the Spanish federation, but more of a sporting organization and not as tied to the government the way CONI is. Thus, the OP paperwork which was held up by Judge Serrano in Spain, when Serrano went on vacation and REFC and others sought the paperwork and got it, CONI ran with the paperwork. Make sense?
WADA isn't part of any judicial action here, and merely set-up rules that cycling federations of nearly all countries abide by.
The UCI is the governing body of cycling. In this situation CONI is going to attempt to pressure the UCI to uphold a suspension of Valverde world wide.
I earlier said this gets messy. What I mean by that is that I don't think there is any set rules we're dealing with here. For example, 7 bags of blood were found to belong to Basso. Basso confessed to "attempting to dope" and was given a near 2-year sentence. However, in the smaller "Oil for Drugs" scandal, Danilo Diluca was found to have intended to dope (without ever testing positive for anything) and was given a 3 month suspension in the off season, by the same authority in Italy. Why? Because OP is a bigger scandal? Because there was actual blood bags of Bassos? Well, with Valverde there is only one bag of plasma with his name attached to it. Whether that plasma had any drugs in it, I can't seem to find out.
I would encourage you to read that link I gave to the Bicycling Forums
which goes into great detail on Operation Puerto and how it pertains to Valverde.
I'll try to post again late tonight, but may end up writing a huge treatise on OP.