I believe so, once an empty slot is filled, a planet can produce no more.
The only way to empty the slot is to trade the resource off.
Of course, some planets have 2 empty slots.
This is correct. Each resource slot can only hold 1 resource at a time, and once produced you need to trade the resource to open up the slot before you can produce on that slot again.
And yes, some of the Fertile planets do have 2 slots, that's kind of their thing... Fertile planets are better at producing stuff.