Ans.1: The sentence is in PPT (Present Perfect Tense). It’s used in lieu of the PST (Past Simple Tense) to replace the past action (‘foretold’) as continuing to the present (the effect). It tells about the past and the present. It talks about the foretelling in the past and its relevance to the present time.
It doesn’t matter whether the prophet is still with us or not. The sentence gives no such indication. Indeed if you use the simple tense, it would have to be in the past. The PPT is fine.
Ans. 2: ‘Signs' is the DO. I suppose you could add ‘we’ to the sentence as an IO or as one of the double objects.
Ans. 3: It’s kind of a shortened form of ‘The holy Prophet has foretold some signs by which he wanted (would like) us to recognize the approach of the Day of Judgment.’ You may also use ‘in order’ in ‘The holy Prophet has foretold some signs in order for us to recognize the approach of the Day of Judgment.’