Hi Kuba
If you rent a house in Green Valley you should look for one with a phone line. Mobile internet is horribly expensive and not suitable for work.
Also, be very discreet about working on the internet as you are unlikely to have working visas in Indonesia. The issue is a grey one, because technically you are working in Indonesia as you are physically there, even though you are produce that work in another place virtually, and are likely employed by a company outside Indonesia and paid outside Indonesia. To my knowledge it hasn't been tested in court, but you do not want to run foul of immigration.
Also, be aware that Lombok suffers from frequent power outages. If you are reliant on power for internet and computer that can be an issue. You may need a generator, or work on laptops with wired internet connection and be prepared to head for the beach or bar when the power goes out.
Green Valley is a series of 'BTN's' or small suburbs and is located near to Senggigi. There is an ok beach across the road from the entrance to that area, with plenty of cafes and several bars in walking distance. There are good and bad areas of Green Valley - the bad ones are full of drugs and prostitutes and you don't want to live there. Before you decide what to do, meet some local expats and ask questions. Good places to meet are Beach Club, The Office, and Happy Cafe (particularly Saturday nights). A visit at night or early hours of the morning to the street of a house you are interested in will also tell you if the area is quiet or not.
There is a small wave for surfing off the point of Senggigi. It's nothing to get excited about. All the good surf is in the south. Nothing to stop you heading south for the weekend and doing some surfing, stay in a cheap hotel, but live in an area with more facilities like the Senggigi area.
Motorbikes - you need a motorbike license to rent a motorbike or scooter. Most start at around 90cc up to 125cc, whether they be a bike or scooter. Scooters are popular as they tend to be more comfortable to ride and are automatic, but are town bikes, so not really suitable if you intend to travel around the island (or out to the surf beaches). You need an international driver permit that is endorsed for a motorbike not a car. So, either get a motorbike license before you leave home, or pay the money and time to get one in Lombok. A car license, even if it allows you to ride a 'scooter' in Eurpoe, is not suitable. You may not be able to get a license in Lombok if you are on a tourist visa.
Make sure you have adequate insurance that will cover a medivac out of Indonesia if required. Consider whether you will take anti-malarials or not. Long term is not really feasible, but be aware there is cerebal malaria around banko banko (the surf spot).
You haven't mentioned visa - so I assume you know the score there.