Toni.  I don't have any particularly useful advice about this topic.  I butt my head up against it often.  

Regarding the use of italics and the often clunky-looking 'he thought' tag, it mostly comes down to clarity.  If it really is clear without the tag, then you should omit it, unless the phrase demands it for reasons of pacing or rhythm.  I use italics (as a way of conveying the actual thoughts of a character) if they are the actual first person thoughts of that character, and actually I don't use them very often. 

I'm going to post an example from my own WIP because it's right in front of me and I'm too lazy to go looking for something better.  It's only a first draft, so bear with me, but it does demonstrate both concepts:

    Something else, another thought, kept crawling its way across her mind:  she wasn't going to die.  She was so used to the idea that death was     immanent that it was hard for her to really believe.  I am not dying.  There was lifting sensation at the bottom of her stomach, an odd,     wriggling lightness.  But she didn't really know what to do with the information.  I, she thought, am not going to die.

We have it three times, the same piece of information repeated for emphasis, because it really is a concept she can't get her mind around:  First presented as a declarative statement in third person;  next stated in italics (because it's a first person statement) but without a 'she thought' tag because it's perfectly obvious who is thinking; and last as a first person statement but with the tag because it adds even more emphasis to the statement by breaking it into pieces, making the reader's brain spend just that extra microsecond reading and processing it.  

As far as your questions about narrative mode, books have been written.  Third person limited really means told from the perspective of one person and one person only.  No hopping from head to head, no omniscient commentary.  Examples are everywhere.  Harry Potter is third person limited.  With very few exceptions, (the first chapter, for instance, when Harry is a baby) if Harry doesn't see it, hear it, think it, etc, then neither does the reader.  (It's worth noting that writers cheat at this all the time.  If a break in perspective doesn't leap out at you as a break, then they've gotten away with it.) 

Omniscient narrative usually means having a narrator in the classic sense, someone who knows everything that's happening (or more accurately, has happened, unless you're writing in present tense.)  If the narrator gives us the thoughts (or the perspective) of a lot of different characters, that's an example of omniscient narrative.  Tolstoy does this in Anna Karenina, just as an example.  Works beautifully, despite the head hopping.  Good writers make the narrative work for them, rather than letting the form cripple them.  The final litmus test is always:  does it work?  Is it clear, is it compelling, does it make you keep reading?  

(A narrator can, of course, also be a character who doesn't have any idea what's going to happen next.  In The Great Gatsby, we see everything through Nick Carraway's POV, but he's hardly omniscient.  (Nor is the story particularly about him, though there's poignance in the way the story effects him.)  Likewise Ishmael, who serves as the reader's surrogate (our man on the Pequod), though Melville let's us see conversations and scenes he couldn't really have seen, but nobody cares at that point.)

Wow.  I go on.  I really do think analyzing this stuff is a good thing.  Keep your books straight (as you would in accountancy)  Know your techniques.  Internalize them.  Make them your own.  Then just write.  If it's good, your reader won't give a rat's ass whether it's third whosiz avuncular clandestine normative, or first personal absentia subjunctive.  

Be clear.  Be compelling.  Tell your story. 

Cheers.

This is great, thanks! Appreciate that you provided your WIP as an example and presented it in different ways. That was very helpful. I really don’t like the tag. If it’s in italics it’s pretty evident who’s thought it is, that is, I believe, as long as it is third person limited.  Love your analysis of the various narrative forms, and thanks for the examples.

Regarding the use of italics and the often clunky-looking 'he thought' tag, it mostly comes down to clarity.  If it really is clear without the tag, then you should omit it, unless the phrase demands it for reasons of pacing or rhythm.