See answers in line:
If I will be using Site Recovery Manager, Do I need to configure distributed switch on the DR site if the VMs from the production site are already using distributed switch. - No, SRM can work with different types of switches at both sites.
If the vCenter Server and the server itself is using its own domain (not joined in the AD/vsphere.local), can I still install SRM and configure it or vCenter Server and its server should be joined to AD? - Yes, SRM doesn't have to be joined to the same AD domain (or a domain) at both sites. I would recommend if using vC 6 to connect both PSCs to the same SSO domain. Keep in mind this is completely separate from AD domains.