Hello,
I was born and grew up in the island.
Maybe I can assist you with general condition of the island ( I cant really suggest which travel agent/guide etc to use in the island, I left the island 1995 although now still have chance few times a year visit my parents there )
1. The best route in terms of money is Singapore-Denpasar-Mataram by Air Asia and Merpati has several flights a day between the two, Garuda only have one flight in the evening. Denpasar-Mataram each way cost around IDR 375,000/pax = S$50 approx.
2. Depends on how good is your physical condition. If not too good I would suggest you to visit Gilis ( Gili Trawangan, Gili Meno and Gili Air, Trawangan is the most populated and tourist center with hotels and guest houses ) You can do snorkling and diving.
This is one of the views from Gili Trawangan ( exclude the lady in the photo of course
)
http://www.flickr.com/photos/kusir_cidomo/5351150867/in/set-72157626158566367/
Also visit Kuta or local says Kute, yes there is Kuta in Lombok not the one in Bali.
You can view my Lombok set here
http://www.flickr.com/photos/kusir_cidomo/sets/72157607784518983/
3. Yes it is safe, no worries. You will see a lot of tourist walking or motorbiking in the island.
4. Nothing much standard precaution for valuable items like passport,wallet etc make sure you keep it safe. And just watch out when you cross the street, many motorbike + cidomo ( local traditional transportation ) there and bring mosquito repellent, sunblock, sun glasses.
Best time to visit May to September when it's dry as you will see a lot of blue sky and clear sea water.
hope that's help.