Can my GameCenter Apple ID be different than the one I use to download apps?

Yes. You can have as many AppleIDs as you wish, and you can use different ones for different Apple services (game center, the stores, iMessage, iCloud, facetime, these forums).

However, note that anything purchased with your existing AppleID remains forever associated with that AppleID. You will not be able to update purchased apps or content without that AppleID, and you can only change your AppleID in the iTunes, App and iBook stores once every 90 days.

With iMessage your AppleID that you sign in with determines the account. If you tap on the send&receive setting, your Appleid will be at the top in blue. Tap it and sign out, then sign in with the new AppleID.

Note with iCloud you will be sharing everything - notes, mail, calendars, contacts, etc. If you want privacy then an alternative is to each use your own AppleID for iCloud as well, and use family sharing to share photos and other content.

How do I then do this? For example, my wife and I want to use the same AppleID for iCloud storage (including photos) and are happy to use the same ID for apps, iTunes, etc. However, we want to have individual IDs for FaceTime and iMessage. I can see in Settings on my iPhone where I might be able to specify a different account for FaceTime, but not for iMessage.

Thank you for your speedy reply. The problem we are trying to solve is that I have a lot of storage available under my iCloud account and, rather than paying for a separate iCloud account for my wife's photos, I would rather they get automatically stored into my account. For issues such as mail, calendars, contacts, I don't use iCloud for these in any case, so this should not be problematic. As long as I can sign into iMessage/FaceTime using a different Apple ID than I am using for iCloud Drive, Photos, etc. then I think this should solve my problems.