I just bought a wired router to hook up my two computers and it is working ok so far except for one disconnect.
I am using AT ans T as my internet service provider and it is a PPPoE type where I have to sign in my username and password each time I want to log onto the internet.

I set the wired router to disconnect after four hours, and apparantly it did, but then I could not reconnect to the internet and so reset the router to
be"alway on," where it would keep checking that the internet was connected every 30 seconds and if the internet did disconnect the router would automatically reconnect for me.
I had no problems initially setting up the router and so did not make any changes that the router did not automatically set up for me (except for giving the router my AT and T username and password.)
But I am concerned about being constantly connected to the internet and whether my computer is safe or not.
Is my computer safe if I am connected to the internet but my computer is turned off ?
If my computer is turned off doesn't that mean that Zone Alarm is off and I am more vulnerable ?
I have altered no settings in Zone Alarm since setting up the router except
for when Zone Alarm noted my new network connection (upon the router being set up) where it asked me whether to
set it for "Trusted"
"Internet." I chose "Trusted" because ZA
is on my two home computers although my computers are not configured for sharing in any way other than through this ZA setting; I am just using the router to share an internet connection, not to share info between the two computers.

So, my question is: is it safe to have a constant internet connection ?

Would it be safer to log off my username and password each time I sign off, and then relog in when I want to log back in (does it make a difference ?)

My ignorance on this is making me very nervous and I feel like my computer is just "out there" with a big ole "Come and Get Me Sign" on it.
Any advice would be appreciated :8}
In case you have not guessed, this whole router thing is totally greek to me :8}

My router is a: Linksys Etherfast Cable/DSL Router with 4 Port Switch.

Operating System:Windows XP Home Edition
Software Version:6.5
Product Name:ZoneAlarm Internet Security Suite