That’s a good question, since entry into the Matrix requires being physically connected with the assistance of someone jacking you in and then disconnecting you. An operator seems to always be necessary except in this instance.
2 & 3. Who knows? The agents don’t seem to know about her, though they have the ability to show up where the rebels are.
They don’t explain it in the film, so we have to speculate. I think they have the ability to leave lines open, but they don’t do it because of security. Since Cypher was working for the bad guys anyway, he didn’t have to worry about them exploiting it. Sort of.
It’s not known in The Matrix whether the Oracle is human or AI.
And in that scene where Neo and Cypher are talking, Cypher turns off 2 monitors as soon as he notices Neo there. And in the very beginning of the movie when Trinity and Cypher are talking, you see a 6(I think) appear and Trinity asks Cypher if he heard anything, and Cypher says no very suspiciously.
I’d just assumed that jacking into the Matrix would be something to pass the time. Cypher doesn’t (as far as we know) do anything on the ship, so he might as well head in there if he’s bored. Whether or not that’s a standard activity for others, we don’t know.
Me, I’d rather mess around in the ‘white space’ with all the junk I could think of and abuse my super powers. (“Tank, load me in an M1 Abrams. Now load in three hundred mimes. Watch this!”)
My own question is where the heck Cypher got that crazy electro-gun. Did he build it himself? Is it standard equipment on the ship?
As to the “goof” Otto quoted: What the heck do they mean “he’d have no reason to deal”? His entire reason to deal is that he’s sick of having to live in the real world!
Regarding question one I always assumed that he left a line open like Achernar suggested, or it was a timed deal, where after he’d been in for an hour or so the hard line would ring enabling him to leave. Still be a bit inconvenient to have to reach around behind your head to unplug yourself in the real world but doable.