As for MX records, these can be pointing to Office 365 or your On Premise Exchange environment, depending on your needs. As long as the Hybrid Configuration Wizard worked correctly, mail flow should be seamless between the two systems.
[PS] C:\Windows\system32>get-hybridconfiguration | fl
RunspaceId : de34fead-a695-444b-44e7-44cc4444becb
ClientAccessServers : {}
TransportServers : {VSVR-EXCH2010}
SecureMailCertificateThumbprint : 4491AE94D3444C229CBCE4439CDF1444CCB44C244
OnPremisesSmartHost : mail.<domain-0>.com
Domains : {domain-a.com, domain-b.com, domain-b.com, autod:domain-0.com}
Features : {FreeBusy, MoveMailbox, Mailtips, MessageTracking, OwaRedirection, OnlineArchive, SecureMail}
ExternalIPAddresses : {<pub IP Address>}
AdminDisplayName :
ExchangeVersion : 0.10 (14.0.100.0)
Name : Hybrid Configuration
DistinguishedName : CN=Hybrid Configuration,CN=Hybrid Configuration,CN=myenm,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=domain-0,DC=com
Identity : Hybrid Configuration
Guid : 3d96cf44-6f44-4744-a044-dd83e946c344
ObjectCategory : domain-0.com/Configuration/Schema/ms-Exch-Coexistence-Relationship
ObjectClass : {top, msExchCoexistenceRelationship}
WhenChanged : 1/5/2018 5:17:03 PM
WhenCreated : 12/7/2017 12:12:37 PM
WhenChangedUTC : 1/6/2018 1:17:03 AM
WhenCreatedUTC : 12/7/2017 8:12:37 PM
OrganizationId :
OriginatingServer : ADDC.domain-o.COM
IsValid : True
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