You're going to want to verify in the Eventlog-forwardingPlugin/Operational log of the WEF client you see an event indicating the client created the subscription successfully (Event ID 100), and no further Event ID 103 unsubscribe events. If the Event Forwarding target subscription manager was deployed with Group Policy, then running gpupdate /force on the WEF client will cause the system to re-evaluate the event subscriptions and you should see new events appear in the Eventlog-forwardingPlugin/Operational log. This is handy for verifying changes you made without waiting for the normal check-in cycle to initiate.
You'll also want to confirm the NETWORK SERVICE account is in the local Event Log Readers group of the client. It is preferred to deploy this using a Group Policy Preference to your WEF clients. For Domain Controllers, the NETWORK SERVICE account would need to be added to the "Builtin/Event Log Readers" domain group. This is the account used to read and send events to the Windows Event Collector server. Another troubleshooting step is to use the wevtutil.exe utility to confirm the Network Service account has channel access to your logs you intend to collect (ex. wevtutil get-log security).
The client will connect to the WEC server and then apply any subscriptions you have given it access to. Verify the subscription permissions include this client or a group the client is a member of. If the client was recently added to this group, then reboot it to refresh its Kerberos token. I have seen clients fail to stay subscribed in some cases, and they would immediately unsubscribe for no apparent reason. Deleting the clients bookmark registry key on the WEC server resolved this. The key can be found at "HKLM:\SOFTWARE\Microsoft\Windows\CurrentVersion\EventCollector\Subscriptions<subscription name>\EventSources" on the WEC server.
This page covers the entire process for setting up event forwarding and provides more information you'll be able to use when taking Event Forwarding to the next level:
https://learn.microsoft.com/en-us/windows/security/threat-protection/use-windows-event-forwarding-to-assist-in-intrusion-detection
Here is another good resource with step-by-step instructions for setting up a source computer initiated subscription.
https://adamtheautomator.com/windows-event-collector/
Regarding your seconds issue of the WEC server not forwarding events for itself, there is an open issue on this posted to the palantir GitHub repository:
https://github.com/palantir/windows-event-forwarding/issues/37