Thigh gap?

A "thigh gap" as you call is has much more to do with how your body is shaped then most people realize. There are many people who can be underweight and not have one, and I have met a few individuals at a normal or above normal weight who do have them.

Running will not do anything on it's own to give you a trait you want to have. Losing fat as a whole is the only way to truly change the appearance of your body. This means you have to eat healthy, exercise often and take good care of yourself for anything to truly change. There is no such thing as a "target exercise". Pushups do not burn fat on your arms anymore then situps burn fat on your stomach. The exercise will tone the area, but if you don't lose overall body fat through good diet then the toning will have little to no effect.

It's fine to have a thigh gap since it doesn't tell you whether you are healthy or not. It's difficult to get it. It also kind of depends on how wide your hips are and probably bone structure. The wider your hips the more likely you are to get a thigh gap. And you don't need to be underweight to get one.

Running doesn't provide an adequate stimulus to cause muscle gain - the loads just aren't heavy enough, the intensity isn't high enough.

A thigh gap is primarily controlled by your hip structure - something you can't change. Some people have the body structure for it, some don't, and losing all the weight in the world won't change that.

It is actually very difficult to obtain a thigh gap, since it is most common in girls with prepubescent bodies. During puberty, women begin to produce more estrogen, thus, leading to wider hips, a bigger bum, and larger breasts. The ONLY way you'd be able to get a thigh gap is to become extremely underweight. This is why women who starve for far too long stop having their period; they are not producing nearly as much estrogen as they need to.

Jogging is a good idea. Get well made, supportive shoes. Look at champion runners to get some idea of your future re thigh gap.