For your 1st question
ALT-T e
is the fastest way I know of to get to the logs
its a one key combination Alt-T and a second key 'e'
For your 2nd question
You can use the Control Box (Top left corner of the PM Window)
You would then go to the log menu item
click on 'on' and then save to start the log
Thats one way.
to be able to right click inside the Query (PM) window you can use this popup script code.
This code goes inside the POPUPS section and under the Query window
Start Log
.Log:/log on $active -f $active $+ .log
This will create a log file for the query window with the same name as the nickname that appears in the query window title bar.