Try landing on the asteroid first. That'll make sure the ship doesn't fly off... as long as you don't have one of your thruster controls actively powering a thruster.
You can! Yes! Just get your ship close to the asteroid and orient the ship so that the asteroid is below you, then use your down thruster to place yourself onto the asteroid surface. You'll then be able to exit the ship and walk over to the ores. Just be careful. There's hardly any gravity at all, so it's easy to float away. But with careful management, that's how to do it.