Not really. There's no ü sound in English, but there is an ö sound, for example in burn. Maybe that's why you hear the two the same?
Could one use "Grüße" as a welcoming/greeting as well, just like "hallo?" For example, if I was to be hosting a party, could I say "Grüße!" as I opened the door to my friends?
I have never heard it used that way, but I´m not 100% sure that you cannot use it as such. I know that in Southern Germany, a rather formal greeting is ¨Grüß Gott¨ (meaning ´´God greet you´´).
Die is the article for the plural, so because Grüße is a plural noun, we use die :)