"Further" means additional or more, and it is used for non-physical things like information, details, or discussion. "Can you give me further details" means asking for more information, which fits the sentence perfectly.
"Farther" also means additional distance, but it is reserved for physical, measurable distance, such as "he walked farther than me." It cannot describe details or information, so it does not fit here. "Faster" describes speed, not quantity, so it does not make sense with "details." "Furthur" is simply a misspelling of "further" and is not a real word.
Since the sentence is asking for more information rather than physical distance, "further" is the right choice, making option 2 correct.