First, artifact digging only becomes a historic event after the policy in Aesthetics tree.
Second, Great Prophets are great people too and thus trigger the historic events. On top of that you NEED to have Mandirs in order to benefit from UA, otherwise enjoy your assassinations (which are not affected by anti-spy buildings). Also various "when GP expended" triggers as beliefs.
I didn't know trying to grab a Halicarnassus was "a ton" of wonders.
1. That is true, but with Arabia, it is pretty much a given that their supported victory condition is best with tradition + aesthetics.
2. Yes. They are GP. But are they completely necessary? I get that you're trying to maximize everything, but it really isn't necessary for Arabia to win, they are just overwhelming in spamming GP to the point where you can end the game as soon as culture victory is allowed (I think the turn limit was like turn 230?) and with overwhelming tourism as well.
3. I see where this is going. Of course, halicarnassus is great, but I haven't grabbed that wonder in such a long time and was still able to come out victorious in every culture victory (which is how I usually win most games because I'm too lazy). Long story short, I don't think it's needed at all. It's very unlikely that you can grab early game wonders faster than AI. It isn't a gamble, and you shouldn't have a strategy revolving around getting an ancient era wonder where the AI has a huge head start before you. But this allows you to grab the later wonders easier as they take a production penalty for each wonder they have. Globe theatre, broadway, leaning tower of pisa, red fort, great wall, louvre, porcelain tower, roman forum, uffizi and hagia sophia (if you have a religion) are all wonders that Arabia wants, but you aren't going to be able to grab each and every one.
And of course, if you want a religion, you have to commit to it in higher difficulties. The AI will always have an easier time because they start with workers for improvements. It's part of the difficulty spike. Try lowering the difficulty if you want everything to go your way.
Edit: And I don't think Mandirs are needed either. Sure, they are good for protecting your great people, but honestly...mastery is what you want most. If you really need growth, asceticism is what you want.