Existing O365 Environment with AD and Introducing Azure AD Connect

[a-s-17]

I have an all cloud email environment in O365 but an on-premises Active Directory environment.  We must introduce Azure AD Connect to get the two environment aligned.  I know what I need to do to soft match the O365 accounts with the on-premise AD accounts.  I also know I need at least an Exchange Management Tools server to management my users that will now be sourced on-premises.  What I do not know is if I need to run the Hybrid Configuration Wizard.  There is absolutely no mail currently on-premises and there never will be.  Do I still need to run HCW in this situation?  Thanks.

[#2]

Azure AD connect is not only for email solution. Since you have AD in on premise and if you want to enable single sign on you need to sync the account to cloud.
Once you made your environment as hybrid you can pass through the authentication allow users to connect on premise and cloud application by using same password.
You can enable MFA for second authentication.

[#3]

@a-s-17 No, you don't need it. 

[#4]

@a-s-17 If all your email objects - shared mailboxes, distribution lists etc - are already in Exchange Online, and you aren't planning to sync any mail-enabled objects from on-premises then you don't need Exchange on-premises.

Essentially, AAD Connect Sync in this case will mainly sync user objects and only mail properties you would need changing for user objects is email aliases which you can do using AD tools by adding proxyAddresses.