Rotev agvaniyot ?
So this is another example of the construct form that we should be getting a little later, right? Literally, "sauce of tomatoes"?
Yes just like "tomato sauce" in English, only here the word order is opposite.
Yes those type of constructs (בקבוק מים/רוטב עגבניות) don't need ה because you aren't talking about some specefic tomatoes or water, but you are describing generally what this bottle or sauce contains.
By the way- the hebrew "R" its like the danish, not strong. so its lost sometime..
Correct me if I'm wrong, but isn't the Hebrew R more of a guttural sound, as opposed to the more labial R in English?
Yes. Hebrew r is [ʁ], a voiced uvular fricative. English r is primarily alveolar (or postalveolar) though.
Why is the r sound in רוטב not pronounced?
Is this the American tomato sauce, or the English tomato sauce (i.e., American ketchup)?
Well, קֶטְשׁוּפּ is used as a loan word, and it seems there is a standard for it in Israel (תֶּ֫קֶן הקטשופ הישראלי), so Heinz was forced to call it מתבל עגבניות tomato spice some time:
Osem ketchup is the best! (Actually, home made is the best!)
Technically an approximant which I think you've described admirably