Will there be any issues running both activation server and KMS?
No, both can be used.
Does the server have to be online for activation to work or is activation done once server is joined on to the domain, and the server can be shut down?
It need to be online, but it support to be offline for some time (+- 7 days), not forever. To quote Microsoft; After activating, the KMS clients try to renew their activation every seven days.
Any tips on moving servers from MAK to AD-based activation?
That part is straighforward. Activate your KMS host with a KMS key for your server. (check your volume licence for the key). After you simply activate the server with a KMS generic key with slmgr.
Like Windows Server 2022 Datacenter would be; slmgr /ipk WX4NM-KYWYW-QJJR4-XV3QB-6VM33. See key there
Using that key make the server look for the DNS record to know where to register. The KMS clients use DNS SRV records (_vlmcs._tcp) to identify the KMS host. You need a minimum number of Windows Server to be activated with the KMS server to have the activation work.
In case of trouble check that article; https://learn.microsoft.com/en-us/windows-server/get-started/activation-troubleshoot-kms-general, it's a well wrote page that explain how to troubleshoot.