You should try to build the warp drive :) it might go a little faster. You can find the resources for it in the asteroid ring around the starter planet
The purple rock is a bit tricky because they don't stack, each piece requires its own inventory slot. There is a key to open your full inventory and see how many slots you have. It might be TAB? If not then just try different keys till you find it. Once you find that inventory screen, you can see how many free slots you have, and dump anything that is not a priority. Then you should be able to collect the purple rocks for the warp engine.
I did exactly what you did and flew to the moon the long way. It would be fine if there was something fun to do along the way, like monsters to fight or a puzzle to solve.
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.