If its any help...
The idea that muscle-turns-to-fat is a myth. Muscle and fat are two different parts of the body with different components. One cannot simply become another in the very same way a cat cannot morph into a dog through giving it the wrong food. If you stop using your muscles, water which once used to fill the muscles does not and the muscles can appear loser and in some cases look a bit saggy. Some in time can end up losing these muscles and saggyness altogether but actually this means much more work when they do take up the exercise again. Fat is a different organ and can and does shrink in size when losing weight, there are different theories on weather or not the cells ever fully vanish or if they just become very small and will refill if you eat more calories then you burn but thats a separate issue. Just don't eat more then your body requires to live (including moving around) and you shouldn't have too much to worry about.
In terms of exercise and running not being an option, swimming is a good option as it uses all your body and is in most cases the least harmful in comparison to impact borne exercises such as running.
Cycling can be a good option but be careful of your knees! Cross trainer machines at the gym can also be good- there are many different types of exercise techniques you can try on these machines, HIIT for example is far easier on a cross trainer then on a treadmill. Get advice as to how to use machines and keep an eye on your posture as incorrect posture can lead to other injuries and other injuries are the last thing you need! When you put your body under pressure, posture is often the first thing to go.
In terms of how long to rest for, this really varies person to person. It depends on your age, your heatlh, your stress levels, nurtition levels...even genes, its not so easy to give a hard and fast rule. What you can do to help is to keep your nutrition optimal- eat decent quality proteins (hard if your a veggie: Google for "complete and incomplete proteins" for more information on that). It may also mean you need to eat slightly more then your body needs otherwise as your trying to build and repair damaged tissue. Rest weel (sleep well, don't over-exercise) and bear in mind once your older then about 30 your rate of recovery will slow down as your body is in natural decline. Once you hit the menopause its even slower.
I'd say a rough rule of thumb is to rest for at least 6 weeks then start back slowly. The first second you feel pain stop and repeat the process, the longer you run with pain the longer you will need to wait for recovery, shinsplints are not nice!
Massage can and does help because a very important part of recovery is the circulation of the area. Sports massage in the area and around the area (the fleshy calf muscle for example) will all help but careful to not over do it as your body will need time to heal rather then just be palpated. Stretching the calf and achillies can also help after exercise and after massage.
RICER: Rest Ice Compression and Elevation followed by Rehabilitation generally works for most things and shin splints are included in that.
