You won't be able to check in early at 5am. Standard check in is at 2pm. If you are lucky, your room might be ready by midday. If you want to guarantee access to a room that early in the morning, pay for an additional night - I did exactly that a month ago - and it was well worth the expense.
No problem at all. We hardly used any taxis though as the shuttle buses and metro were great. We did do our shopping in Dubai Mall and were charged AED50 for a taxi back to the hotel in a cream taxi. The white Mercedes taxis were expensive.