Is Neo living in two worlds inside the Matrix?

Neo: You ever have that feeling where you're not sure if you're awake or still dreaming?

This line is specifically pointing out that Neo doesn't fit in the Matrix. He's one of the one percent (no pun intended) that don't accept the Matrix as presented.

The Architect: As I was saying, she [The Oracle] merely stumbled upon a solution where nearly 99% of all test subjects accepted the program as long as they were given a choice, even if they were only aware of the choice at a near-unconscious level.
-The Matrix Reloaded

Some realize they are in the Matrix, sub-consciously, while others, like Neo, start to see this at a much more conscious level. The Potential Red-Pills.

As to his jumping from one "life" to the other, there is a few reasons for it.

1. The simplest reason is that Movies tend not to show the mundane things like Neo leaving the club, the train ride, getting home and falling asleep unless there is a pressing need to. Here, it actually makes more sense not to be cause of:

2. It's thematic. The director wants the audience to feel that Neo sees his public, law friendly persona as different from his private, unlawful one. The straight cut between scenes provides the stark contrast that Neo feels.

3. Neo's experiencing Existentialism thoughts. He's clearly not happy about his public life. Existential crisis, doubt about his future. Uncertainty about his goals. Hence his' boss' lecture about his lateness.

4. Jumping from scene to scene is pretty much how Dreams go in real life. You don't always remember the mundane parts, going to sleep, walking to places, etc. Exhibits signs of False Awakening. This is part of Neo's dilemma. He doesn't know if he's awake or not. If he's dreaming. Signs of Lucid and Pre-Lucid dreaming, as Neo is a potential Red-Pill at this point.

5. The final reason is a bit more complicated. Given the extended universe of the Matrix, as provided by books and the Animatrix, there is signs that life in the Matrix isn't always linear, or in real time. It can be faster, or slower. The humans can experience multiple life times.

In "Goliath", the short story by Neil Gaiman commissioned by Warner Brothers for the Matrix Augmented Reality website WhatIsTheMatrix.com (now defunct), the protagonist character is a human blue-pill in the Matrix. Due to outside interference, his server is disconnected from the matrix, and the machines reboot it. The protagonist experiences three or four different life times, and jumps around, not really experiencing everything, but at the same time he does. At the end, they plug him back into the Matrix, and use his previous "lives" to provide him a better life before he dies. He experiences years in real time seconds.

Of course, As the movies show though, everything in the Matrix seems to occur in real time, especially when the rebels have hacked in, and are talking from the Matrix phones to the Operator in the Real. So Goliath seems to be a special case, but it makes it possible.

Thomas Anderson isn't leading two separate lives (in a literal sense), he's simply burning the candle at both ends and suffering the consequences of trying to reconcile getting up early to go to his day-job at Meta Cortex with staying up late chatting on hacker boards. The inevitable result is sleep deprivation and a poor work ethic with persistently over-sleeping and total lack of enthusiasm.

On top of that, he has the added complication that in his "normal" life he's persistently beset by the feeling that something is inherently wrong with the world. That's what's driving him to stay up late chatting on hacker boards and running searches, trying to contact someone who can answer to the mysterious and all consuming question of The Matrix and what.it.is.

When he finally makes contact with Trinity, she approaches him when she can see that he's nearly passed out from exhaustion. She hacks into his computer and chats with him for a few minutes (as he nods along) and when he's just on the verge of nodding off, a Zionese rebel (DuJour) knocks on his door and invites him along to the club.

No.

Most clever outlaws have two lives. One is the "normal" life where you go to work and you kiss your wife and you pay all your taxes on time.

An example of a "normal" life is Mister Brooks when he is working at the office during the day building boxes.

There is a second life. Let's call this life your "criminal" life where Mister Brooks gets out of bed in the middle of the night and he leaves his house to murder somebody.

Another example of a "normal" life is when Vic Mackey is chasing and arresting the bad guys during the day.

He is also selling drugs and doing all sorts of criminal activities including murder after his shift at the police station is over.

He is a crooked cop.

This is better explained in the scene when Thomas A. Anderson is arrested for the very first time by what seems to be an FBI agent.

Here is the scene:

Most software developers (not so much in today's world) used to have the skills of a hacker; he most definitely was a hacker during the night and a real-job guy in the day. Hacking is a hobby, a skill, a buzz to him, and that day he did actually go to the club after he saw the rabbit sign again. He always knew something was up.

As an expert computer programmer/coder he could see the matrix was like a smokescreen! I don't think there was any deeper meaning. He has to earn a living somehow and hacking is at the end of the day a skill-set so I bet he would be a great software developer and therefore able to fund his other interests - which he only has time for in the night time!