I have lost weight through cycling alone - about a stone over 18 months, and that it without making any adjustments to my diet at all. But as others have said, you do need to be cycling hard to see that sort of calorie loss. An hour's vigorous cycling - so you are breathing hard and your pulse is significantly elevated will burn around 700 calories.
As to whether it is compatible with a bad back and whether it strengthens your core muscles depends on what sort of bike you have and whether it is the right size and dimensions for your body. You can get a professional 'bike fit' at a specialised bike shop. They will video you whilst you are on a bike and will make adjustments to the seat, and handlebars so that you are putting the least strain on your body as possible. These sessions are expensive, though, usually around £200, but they will give you a list of measurements that you can then use for ever more to set up a bike.
A more affordable option is to buy a bike from a local specialised bike shop, and they will help you find the right sort of bike for you, and will make sure that you have the seat at the right height etc.
If you are on a bike that is right for you, then you should not get skeletal pain in your back. I sometimes get a tightness in my lower back when I have been doing a lot of hill work, but it goes pretty quickly after some stretches.
I would disagree with Nutella about your core muscles. I found that mine have strengthened considerably from cycling. I used to get pains in my back if I stood up for too long - eg I would be uncomfortable having walked around an art gallery, but after even about 6 months of cycling I found that my core was so much stronger that this didn't affect me any more.
It may depend on what sort of bike you ride - I would imagine that a true road bike requires you to support yourself far more with your core than a tourer does, and that doing intensive hill work and speed work also engages it far more than a steady pace.
I am well impressed by your round the world tour though Nutella - that sounds amazing!