If you are on one ticket all the way to ZAG, then your bags should be checked all the way through. 2 connections does increase the chance of lost baggage issues but there isn’t anything you can do about it, so there’s no point in worrying. You are far more likely to have no problems than to have a problem.

I haven’t transited through FRA, but would assume you can stay airside as you are non-Schengen (SIN) to non-Schengen (ZAG). I’m sure someone with more knowledge than me will come along soon.