Whether we must export the certificate(s) from 2016 and import to 2019 manually, or does HCW handle using the current cert on the 2019 box
You may need to import the certificates to Ex2019 as HCW needs a third-party certificate to work. (self-signed certificate won't be trusted)
Whether HCW pulls in prior receive connector or any other useful settings from EX2016 to EX2019
HCW will configure it for you if you select the new EX2019 server during configuration.
Whether we need to re-run the separate AADConnect setup again after the hybrid host changes
No.There is no need to re-run AADConnect.
Which MS entity handles support? Exchange Online support does not include Hybrid questions (even though we are only using a Hybrid box for EXO relay and cloud mailbox connectivity). Since we don't host on-prem Exchange, that team also does not handle support.
It may depend on what kind of issue you are actually dealing with, and the cause of this issue is on which side.
Here is also a thread from Microsoft Q&A forum which may be helpful:
On-perm Exchange Upgrade when hybrid setup already done with O365
In addition, have you considered installing EX2019 management tools instead of an EX2019 server?
It can also keep you in a supported situation for mailboxes management.
(If SMTP relay is not that necessary...)
More detailed are introduced in the following Exchange blogs:
Released: 2022 H1 Cumulative Updates for Exchange Server
(refer to the Exchange Management Tools Update part)
Removing Your Last Exchange Server FAQ