Troubleshooting: Kitchen Sink Vegetable Soup
To address your points in order:


If you only cooked your soup for ten minutes, you didn't give it enough time for the flavours to 'marry'. A gentle simmer for half an hour would give you better results. A night in the fridge to really let everything blend would be even better.
You will not get crispy potatoes in a soup - the liquid will see to that. The only way you will achieve a crispy potato is by cooking and sauteeing them entirely separately and adding them at the end as a kind of crouton
Green beans don't need much cooking at all - they should still have some bite. If you really want to keep the colour specifically, blanch them separately, plunge them into ice water, then drain and add to the soup for the final 5 minutes of cooking or heating.
Tomatoes help counteract this by lending some acidity and fruitiness. If you don't want to use tomatoes, try a little white wine or cider vinegar to give a bit of zing.