Can client update its DNS resource record which is created by DHCP server

Hi Experts,

We have DHCP server which is NOT on DC. We have configured this DHCP server to update the DNS resource record on Behalf of client with a user account credential. (This user account is specially created for DHCP to regsister the DNS records). 

Ex -

When the DHCP provides IP to a client (Machine1.test.com) and registers DNS record with the user acccount on behalf of client.Then

Que 1 - Can Client machine update its own resource record (Machine1.test.com) when the machiens gets restarted or renews Ip etc..

Que 2 - when the clints renews the IP address, Who updates the record (machine1.test.com).  DHCP server or Client machine ?

Que 3 - If the first DHCP server who registered the record fails (shutdowns due to some reason) Then can the 2 nd DNS server with the sasme user accounts updates the resource record ?

I have gone through the articles on Microsoft Technet about dynamic updates and come up with 3 question. Could anyone please give answer for this 3 question.

Thanks