Is there a way to check if an account is unlocked?
No.
if I use wget to send the unlock request, how long will it stay unlocked?
Until u send "lock" request.
so, how does the network know if an account is online?
I understand that starting a nxt client and opening my account in a web browser(online) will have a chance to earn the transaction fees.
Am I confused about UNLOCK and ONLINE?
If I unlock my account and quit the browser(without locking my account), do I have the chance to get trasaction fees?
If I stop the nxt client further, do I have the chance to get transaction fees?