Will they rejoice in Hashem's name the entire day?
Radak: It will be when they mention His name, for they are helped through it.
Malbim: Hashem's name refers to the fixed natural conduct, which never changes - "Hashem Shimcha l'Olam Hashem Zichrecha l'Dor va'Dor" (135:13). Zecher is an act that comes sporadically; they recall Him via a miracle that is not fixed; it is based on the generation.
Why does it mention Gil, and not Simchah?
Malbim: Simchah is constant joy. Gil is something that comes new, e.g. finding something, good tidings
Why does it say "uv'Tzidkasecha Yarumu"?
Malbim: Tzedakah from Hashem is what He does because He is G-d, without looking at people's deeds and preparation. It has no limit, just like Hashem has no limit.
