Do I need to keep signing in to iTunes & App Store in order to receive and send iMessages in Messages?
No, you just need to be logged into Messages
If so, how to avoid automatic downloads that may require payment?
Whether or not you are logged into an account in the store won't make any difference to some charges e.g. auto-renewing subscriptions (that you haven't cancelled) will be charged even if you aren't logged in (or if you've deleted the app that the subscription is in), and similarly pre-orders if they become available.