You don't strictly need it, but it helps make sure people understand you're making a whole sentence, because otherwise it wouldn't be the default understanding:
עגבנייה זה לא תות = A tomato is not a strawberry עגבנייה לא תות =A tomato, not a strawberry
Because זאת only means "this". זה can mean "this" (male and singular) but can also be used as a copula, and then it is used with anything, male, female, plural or single. You could use a gender and number specific copula, עגבניה היא לא תות.
I don't completely agree, "זאת" can be used as a copula, just not in this case. For example "עגבנייה זאת לא דוגמה טובה" (A tomato is not a good example).
I think it doesn't fit here because "תות" is a masculine noun.
You're right, I modify my previous statement. זה/זאת can be used as copulas, the agreement is with the second noun. הוא/היא - agreement with the first noun.
Does תות mean mulberry or strawberry or either? The translation the came up said "mulberry".
Both: mulberry = תות עץ
strawberry = תות שדה
But when you use just תות, it's usually strawberry. For mulberry you'd use the full name, תות עץ.
How would you say "this is a tomato not a strawberry" and "This tomato is not a strawberry"? I'm guessing the first is "זה עגבנייה לא תות". What would the second one be?
this is a tomato not a strawberry = זאת עגבניה לא תות
This tomato is not a strawberry = העגבניה הזאת היא לא תות
Is tomato spelled with one י or two יי? I have tried it both ways and keep getting told I used the wrong word or have a typo.
Either way is fine, since the rules of spelling without nikkud are not very strict in real life, and both versions are rather common. According to the Hebrew Academy it should be עגבנייה (one 'י for the 'i' vowel and one 'י for the 'y' consonant).