Hi there,
A taxi from the airport to the bus station should cost approx 40EG and can depend on how hard you negotiate. You go to the East Delta bus station at Abbassia... ask for Abbassia bus station, all taxi drivers know where it is and East Delta is the company that runs to Sinai. Make sure you get the full visa in Cairo airport when you arrive (it is the standard one they give you - you don't have to ask for anything extra) so that you can go everywhere and back. It is valid for 3 - 6 months and is essentially a multiple entry visa.
You can not spear fish in Dahab or (as far as I am aware) anywhere on the Sinai coast - it is a marine protectorate for the most part. Best places for shore dives are Dahab and if you can get to it, Marsa Alam. Sharm el Sheik is great for boat dives although boat safaris are also available from Dahab including trips to the Thistlegorm and occasional liveaboards.
You can't get to Petra and back in 1 day. It will take 1 day to get there and 1 more to get back and you need at least 1 day to visit Petra itself. I organised a Petra tour for myself and 2 friends on our last holiday with Halabia Tours in Aqaba with a guy called Ziad with that itinerary and it cost us US$265 each including ferry tickets, entry fees, visas, departure tax, all transport and accommodation at half board which I thought was a pretty good deal myself! I am sure there are numerous agents in Aqaba that will do the same. You can also organise it from Dahab but I found the prices a bit more expensive.
The only thing I can think of to be careful of is your patience and bargaining skills. Things take a long time in Egypt and being patient and polite gets you far. Having to barter for everything can be tiring but it is also fun if you give it a chance. Other than that it is one of the most pleasant countries I have ever lived and travelled in.
Have fun!!