Using multiple devices with the same cloud accounts I bought a new phone and everything is now up and running, and the old one has been disconnected from the Internet since I set up the new one. Now I'm considering using it as a backup phone for situations where I could lose or break down the new one.

Is it ok to connect both phones to the Internet at the same time or will that mess things up? I'm thinking of problems like different versions of an application writing configuration info or metadata in different formats, and messing up one another.
It is very unlikely that such a setup messes anything up. The only possible scenario where this could happen would be a badly coded app.

In most cases, your apps would only allow you to log in from one device. Once you login on another device, you get logged out on the first. For instance, WeChat works like this.

Other chat clients like Skype or Google Hangouts push your messages to all other devices that are connected. Email clients should work in the same manner.

Cloud storage apps like Dropbox or Owncloud synchronise your files. Unlike messaging apps, they allow modifying and sharing of files. In rare cases, synchronising errors can occur. To solve such conflicts, the cloud server creates a duplicate. This would only happen if you try to edit a file from two or more devices at the same time.

Apps that include an editor (such as Google Sheets or Simplenote usually allow only one version at the same time. That means, even if more than one person edits the file, there won't be any synchronising issues as changes are made instantly. If these persons delete each others changes, data loss would also be instant and can be reversed using the history function of the app.