You have to get out of your ship. You can "EVA" for a minute or two, just to give you enough time to fly manually over to the rock, mine it, and get back into your spaceship. Highly recommend saving your game before you try.
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.