Hi there ResmeN,
will I have trouble getting a taxi from Bangasai to the the airport if I don't book in advance?
Usually you do need to call a taxi, but they are not far away. You do have to walk about 5-10 mins up to the gate. I recommend you download the bluebird taxi app, it works well.
You can book a through ticket in Gilis that includes the public boat and shared car to Kuta that should cost about Rp150,000 per person. If you make your own way to Bangsal a private car to Kuta should cost no more than Rp400k.
Public ferry + bluebird taxi all up will take around 3 hours from Gili T to Praya airport.
If you take a public ferry between 7-8am then take Bluebird taxi from Bangsal just after the boomgate you can easily make the 1pm flight as you'll arrive to the airport around 10-11
They're sold all over the island. Read the boards outside the shops and go with the cheapest one. It's a van