I was wondering if a single machine (with several network interfaces) can be, at the same time, a DHCP server and a DHCP client, that is the server would dynamically attribute an IP address to an other interface ?
Cetainly it's possible, and indeed moderately common for small routers linking to ISPs.
The external interface to the ISP gets a dynamic address from the ISP (by LCP or DHCP as appropriate), and very often is a DHCP server for the local clients on some internal interface.
This is a portion of a Cisco router configuration with exactly that functionality:
ip dhcp pool 1 network 192.168.241.0 255.255.255.0 default-router 192.168.241.1 dns-server 22.214.171.124 interface Vlan1000 ip address dhcp ip nat outside