Yes, at least 'téged', since it stands in a comparison with 'nem őt'. 'Én' can be omitted, unless you want to emphasize that too.
I know that 'see' and 'watch' are 'lát and 'néz', respectively in many situations.
például: "Most nézem egy filmet" (I am watching a movie) a few days later you can tell someone: "Már láttam" (I have seen it already)
I don't know (and would be curious) how 'look' fits into the picture here.
Néz: it's a deliberate action, you do it with your eyes. You decide to look at something, so you watch it, you stare at it for a long time, etc.
Lát: it's an unconscious action, you do it with your brain, through your eyes. You see, because you have two good eyes, and your brain is able to process the pictures in front of you.
Nézem, de nem látom. - It means that I am looking at it, I am studying the details, but it doesn't make any sense for me, because I struggle to put meaning behind what is before my eyes. It is also an expression for implying that I know what's in front of me, but I don't understand it.
With your example of a movie:
Most nézem a filmet. - I am sitting in the cinema and looking at the screen. It is only pictures and movements and words right now.
Már láttam a filmet. - I have seen the movie, I can tell you what it was about, or at least I can tell you my opinion. So, there is a meaning behind the pictures now.
I'm not sure about that. You can look at a painting for hours. But my question was more about whether lát or néz should be used, or something else. And how I can tell which one should be used.
I think watching is focused on movements or actions, looking for checking the appearance (the look) of something, see is for fulfilled observations, I think.
Look at that child. Watch how it climbs up the ladder. Have you seen it?
But what should be used in Hungarian? Are there (un)written rules when to use what?
Watch can also be translated as 'figyel', as in pay attention. Figyel can be used as guard and control.
Figyelek az autódra/figyelem az autódat means that I am watching your car, and I will stop anyone who tries to steal it.
Ránézek az autódra means that I am checking if your car is there. Maybe it ges stolen between two glances.
Nézem az autódat means that I am watching your car, admiring its beauty, and I may witness if it gets stolen, but that's not my business, I won't stop them