In Windows, when I select static IP (instead of DHCP-assigned one) I have to enter the IP address for DNS servers. Is there a way to manually set IP address yet obtain the DNS addresses from the DHCP server?
No, this is not possible. You can either use static or dynamic configuration, not both.
However most DHCP server allow you to assign an IP address to a specific MAC address. This way the client gets the IP (and things like DNS server etc.) from the DHCP server but it will always be the same. You can set this in the configuration of you DHCP server.
You can tell your router to give that computer a "static lease" which is to say, to assign it an IP dynamically, but always the same IP (it binds to Mac address usually). Most DHCP setups allow for that.
With every residential-class router I have used, you can point the client DNS to the router's IP and it will forward DNS requests to whatever DNS IP the router is using. The end result is the same.