1) SCOTUS allows groups, organizations and individuals to submit amicus briefs in support of or against the petitioner. It happens in pretty much every case before the court. IMO, they are rarely swayed one way or the other by amicus briefs, but they may provide a viewpoint from which the justices or their clerks have not considered the case.
2) In cases against states that have bearing on federal law (which Snyder certainly does, in many facets) SCOTUS will usually invite the solicitor general’s office to file an opinion. This case is not unusual in that regard, and it should not be taken as anything relevant to the case being specifically about sex offender registries.
3) Although I think most would agree that the SG’s office is taking a position we didn’t see coming, it is clear that their opposition to SCOTUS hearing the case is a strategic move intended to keep the registries in place. I personally feel that the feds are starting to see the writing on the wall and that SCOTUS could very well rule against MIchigan’s scheme and in effect render registries in their current form very dangerously close to unconstitutional. I still believe the justices will opt to take up the case. If the court make-up stays as it is through next term, I can see a 5-4 victory for the Does, and everyone on the registry.