Honestly, you've been given so much different advice, and have tried so many different things, the chances are everything is a bit of a dog's breakfast now
In your shoes here's what I'd probably do:
1. make sure your /etc/network/interfaces file contains ONLY the loopback (lo) definition, i.e.
2. stop both network services (one or other may already be stopped so you may get a message 'stop: Unknown instance:' )
$ cat /etc/network/interfaces
# This file describes the network interfaces available on your system
# and how to activate them. For more information, see interfaces(5).
# The loopback network interface
iface lo inet loopback
3. remove any non-default connection profiles from network-manager
sudo service networking stop
sudo service network-manager stop
4. make sure the computer is connected to the modem and then restart the network-manager service
sudo rm -f /etc/NetworkManager/system-connections/*
This should at least get you back to square one (a single, default wired interface) - from there we can try to figure out why the interface / DHCP is not coming up correctly - do NOT edit the connection at this point, nm is pretty good at setting up a default wired connection without manual intervention
sudo service network-manager start
If you want to get down into the muck and bullets you could then try this:
5. start a tcpdump monitor session looking at your wired interface for traffic on the DHCP ports
(it will appear to hang - just leave it open in the terminal)
sudo tcpdump -i eth0 -vvv -s 1500 '(port 67 or port 68)'
6. go to the nm-applet and select 'Disconnect' - and then try to reconnect to the default wired interface (probably called 'Wired interface 1')
You *should* see (in the tcpdump terminal) some DHCP negotiation packets being exchanged between the computer and the modem's DHCP server (if it has one) as it tries to reconnect