Technicolor TG582n diagnostics - ping

[#1]

The techicolour is not going to respond to pings on anything connected via a LAN port that are on a different subnet with a standard configuration. It might work of you change it's subnet mask to 255.255.0.0.

 Of course. I have devices on both subnets. I can ping the technicolor on its subnet from most devices on that subnet. I can't ping from the firewall/router to the technicolor.

I wanted to know if there was a way of pinging from the technicolor to the firewall, but it appears there is not.

[#2]

Again not being a network expert, if you want devices to communicate on different subnets, don't you then have to mess around with default gateways? or as @jelv says change the mask.

[#3]

Well, we're not really at the stage of being able to communicate between the two subnet/LANs, because we can't ping the firewall/router from the 'outer' LAN.

What should happen is that stuff on the 'outer' LAN can't communicate with the 'inner' LAN because it doesn't know where it is, and neither does the technicolor (and it is the default gateway for the 'outer' LAN). The 'inner' LAN can find devices on the 'outer' LAN because its default gateway is the firewall/router, and the firewall/router 'knows' where the 'outer' LAN is.