After very long wait, Apple finally released an iCloud update (v5.1) that works with Outlook 2016. The iTunes updater should offer it to you, if not or if you don't have iTunes installed, you can download it from Download iCloud for Windows. iCloud now works with both Outlook 2016 and Windows 10.
If the 'you need to repair the install' error comes up the first time you run iCloud, close Outlook and sign out of iCloud. The error should be gone when you sign back in.
If the iCloud is not syncing with Outlook, open the iCloud control panel and sign out, then sign back in. If you are asked if you want to backup the calendar and contacts locally, choose No. This in only needed if you are going to stop using the iCloud.
If none of the above works, verify the iCloud addin is enabled in File, Options, Addins. Signing in and out of the iCloud should enable the addin, but there could be something else preventing it from loading.
If iCloud won't stay enabled, you'll need to edit the registry. Browse to HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Office\Outlook\Addins\Apple.DAV.Addin. (If it's not listed in that key, look at the same location under HKEY_CURRENT_USER.) The LoadBehavior value should be set to 3. double click on it to open it and change the value if needed.
If you need to keep enabling the addin, see Always Load an Outlook Addin for the registry keys needed to force it to always load.