Using Outlook for Android for personal email and calendar management
It sounds like your contacts are successfully syncing to your phone, but the Outlook and Gmail apps are not using that contact source when you try to add recipients to an email. Since changes made in Desktop Outlook appear on your phone’s Contacts app, the Outlook.com sync itself is probably working.
First, check that the contacts are actually stored in your Outlook.com account and not just appearing through a local phone sync. Open your phone’s Contacts app, select one of the synced contacts, and check which account it belongs to. It should show your Outlook.com/Microsoft account.
Next, check Android’s account sync settings. Go to Settings → Accounts → Outlook.com/Microsoft account and make sure Contacts sync is enabled. Tap Sync now to force an update.
In the Outlook mobile app, open your profile picture, go to Settings, select your Outlook.com account, and confirm that contact syncing is enabled. Also verify that Outlook has permission to access Contacts in your phone’s app permissions.
Because you have several accounts (Gmail, IMAP work accounts, and Outlook.com), Outlook may not be searching the Outlook.com contacts when you compose a message. Try creating a new email, tapping the To field, and searching for a contact by name or email address. If available, choose the People/Directory search option.
Your method of setting the Outlook.com OST file as the default data file can work, but it is an older workaround. The more reliable approach is to let Outlook.com act as the main Exchange account that stores your Contacts, Calendar, and Tasks in the cloud. The OST file is only a local cached copy, so the important thing is that the contacts exist in the Outlook.com mailbox.
A useful test is to sign in to Outlook.com in a browser and open People/Contacts. If your contacts are there, the cloud copy is correct and the issue is with phone contact indexing or app integration. If they are not there, your Desktop Outlook contacts may still be stored locally rather than in Outlook.com.