The logout time is 30 minutes. No there is no place where you can manually set your own logout time. Yes, the logout time can be increased for everyone, but that would not be a good idea. Lawnsite at times has 250+ members signed on at once, with 600+ guests. Increasing the time would just put more load on the server.
What I don't understand is why you can't copy and paste your reply in at a later time. I do it all the time, but not for the same reason as you. Many times i get distracted, have to leave the computer to tend to something at the house... so I right click and highlight what I have typed, then click on copy. I leave the computer, come back an hour later, find the thread I was interested in responding to, then paste in my reply. It works every time.
So if you have been typing awhile, just to be safe, highlight and copy your text. If you get the screen where it says you must log in, then re-log in, and go back to the thread you were replying to, then paste in your reply.
Other than what I just typed, i'm unclear how to help you, as I don't understand why your reply won't show up if you copied and pasted it correctly.