Q. In the event of a failover to Azure can I keep the same IP addresses?

A. It depends. You cannot have overlapping IP spaces between on-premises and Azure as this would block routing between them as the same IP addresses would exist on-premises and in Azure however in a disaster scenario the assumption is the on-premises network no longer exists which means that network space could now be used in Azure. This means technically you could take the IP space used on-premises and create a virtual network in Azure during a failover that uses the same IP space which means VMs could keep the same IP addresses. This is made possible as through BGP the new path to the IP space would be published and used by other connected networks to find the VMs network even though it had now moved. With BGP its actually possible to move a subset of a network as the most specific network takes precedence, for example imagine you have 10.1.0.0/16 on-premises and then in Azure advertised 10.1.1.0/24 this is more specific and so traffic to 10.1.1.0/24 would go to Azure while the rest of 10.1.0.0/16 would go on-premises. You can even advertise via BGP specific IP addresses using a /32 address.

Note where possible avoid hardcoding IP addresses and instead leverage DNS which can then be updated to point to a different IP.