DHCP Server not Integrated in AD Best practices

DHCP isn't ever "integrated" in AD. DNS zones are integrated at the forest or domain level, and DHCP is used to assign the AD Integrated DNS servers to client machines. So your setup is correct if it follows that standard.

As for the DNS server not being updated, look at your scavenging settings: https://social.technet.microsoft.com/wiki/contents/articles/21724.how-dns-aging-and-scavenging-works.aspx goes over scavenging in DNS. Basically, DNS scavenging is used to clean up old, obsolete DNS entries that have passed a certain age threshold. Note that manually assigned DNS records are not subject to scavenging and have to be cleaned up manually if they no longer apply.

TL;DR - You don't need to have the internal server names on your certificate. Only the public DNS name for the gateway, with that same name in DNS pointing to the connection broker.

When you get a DHCP address from a server that is an authorized Windows DHCP, the DNS is updated with the host names even when the client isn't pointing to DNS server. I believe this is what OP is referring to. So, no, you cannot get this with a standalone DHCP

ensure scavenging is enabled on any one DC for domain.com AD integrated zone

Also set dhcp credentials (standard domain account) under IPV4 scope properties \ credentials tab
Add your account used for dhcp credentials to dnsupdateproxy group on DC
Finally set dhcp dns integration on dhcp server (ipv4) properties with 1st option selected (dynamically update host and ptr records only if requested by dhcp clients
In this case clients are responsible for updating dns host records and dhcp will update ptr records
Another setting also work

Question Mahesh

New Input the DHCP is a Server holding Many Scope and the leases times of the different Scopes  

Question 1: Is there a relation between the days in the scavenging Configuration with  DHCP leases time?

Question 2: What about the Servers that are not registered trough DHCP . activating the scavenging will not wipe those entries ?

Ideally Scavenging duration should be half of dhcp scope duration but you should avoid keeping too small scavenging duration (shouldn't be less than 3 days as per my experience) , else it may delete domain controller records

Server with static ip should update dns records time stamp once in 24 to 48 hours