Paying by card in Bali is a fairly safe option. Most of the bigger restaurants, resorts, shopping stalls will all accept payment via card. Taxis, small local restaurants (warungs), beach bars, and market stalls still only accept cash.
Well, I might say I always use Grab application, never disappoint me so far. However, Grab is banned in some certain locations in Bali, so I'd suggest you to call to the driver first and organize the safest meeting point. In addition, I agree that Blue Bird is a reliable company in Bali for taxi.
The $1000 will be enough, but does that include VOA and departure tax?. On your intended plan, I expect you will spend $60-$70 a day however add $13/day to cover VOA and departure tax. Waterbom will cost extra on the above so you won't have a lot for shoping, which you will want to do..
We recommend budgeting $630 for a week in Bali. This will cover you for private transport, a few restaurant meals and a tour or two to your favourite attractions.