Yes to question 1; there's an option to change an IP address (which has been assigned by the DHCP server) and to also fix that IP address so it doesn't change.
Additionally; lease times can be set anywhere between 1 hour and 21 days.
All settings are in the advanced section of the web-interface.
vDSL2 FTTC Infinity with BT
DL Sync 80Mbps
UL Sync 20Mbps