There are a fair few models out there, and giving answers with respect to each model would get a bit over-elaborate. I'll give the detailed answers for the DE flat Earth model, with mention of others.
1. All forms of man-made satellites are typically just in-atmosphere equivalents, like planes or balloons, if they're acknowledged to exist. Some point out the difficulty with observing something as small and fast as the ISS in any detail, that they deny it being any more than a pinprick of light, which could be anything.
2. Gravity could exist on an infinite flat plane, which is a model some accept. More typically, though, FEers believe gravity does't exist, or if it does it's much weaker or is only exerted by some forms of matter. There are various alternative explanations for what keeps us on the Earth's surface; under DET, for instance, there's what's essentially a flow of force to the centre of the disc-shaped Earth (DET essentially involves two discs on top of each other; the gap between them is what attracts this force, long story). As long as you're above either disc, you'll end up pushed to the Earth's surface.
3. I don't know the more general answers to this. Under DET, though, there's a superheated metal ball that exists in the centre of the Earth (created by the aforementioned flow inwards, which made the matter condense and created it in the middle) which spins quickly, creating the magnetic field in a similar way to a dynamo.
4. Typically no. DET's the only exception I've seen where there's any meaningful core to the Earth. That being said, there's a user called Sceptimatic who has a model with an object in the North Pole which isn't exactly a core, but it's in the centre and has some related traits, though I don't understand that model enough to say much else about it.