For keeping the connection alive, you can check in 
/etc/ssh/ssh_config the line where it says ServerAliveInterval, 
that tells you how often (in seconds) your computer is gonna 
send a null packet to keep the connection alive. If you have a 
0 in there that indicates that your computer is not trying to 
keep the connection alive (it is disabled), otherwise it tells 
you how often (in seconds) it is sending the aforementioned 
packet. Try putting in 120 or 240, if it is still killing your 
connection, you can go lower, maybe to 5, if with that number 
it doesn't happen, maybe it is your router who is dumping the 
connection to free memory.