Can I sign in with one Apple ID for the App Store and a different ID for Apple Music?
The "Mac App Store" accessed through "App Store.app" uses a totally seperate login system from iTunes.app, so you can use different AppleIDs for the two of them.

Within iTunes.app, I use multiple different AppleIDs to purchase iOS apps in different regions, which means I sometimes need to sign out of one AppleID and sign into a different one when switching regions and purchaseing or updating iOS apps. I have not encountered any issues when pushing these apps to my iOS devices or when running them on the iOS devices.

The Apple Music part is trickier.  According to http://www.idownloadblog.com/2015/10/04/apple-music-foreign-japan-account-in-usa/ you cannot easily have multiple regions using multiple iCloud Music Libraries, swithching AppleIDs for this purpose is limited.

As I understand things: if you want to use a different AppleID to buy some iOS apps, this should not be a problem - your current setup will not be impacted. If you want to use a different AppleID to sign into Apple Music then you will loose access to your current AppleID Apple Music stuff until you switch back.