The comments (above) answer your question with great analytical exactitude. I offer a lower-brow account (by necessity). To start with, both "After cooking our dinner we would remove" and "After cooking our dinner we removed" are correct ways of describing certain actions; tht is to say, neither wording is wrong when set up properly and when put to work to describe the appropriate event or events you have in mind.
The version with "would remove" works better when, as John Lawler puts it, you are "describing repetitive or habitual actions." For example, if each evening for several weeks weeks during one memorable summer you regularly removed the middle seat after cooking your dinner, the choice "we would remove" is a somewhat better option than "we removed" for conveying the repetitive nature of the action. On the other hand, if you're describing a memorable one-time-only action, you may do better to avoid the implication of multiple instances that "we would remove" introduces and instead use the one-off-friendly option "we removed."
To this point my answer has assumed that you don't indicate elsewhere in the surrounding text whether the seat removal was a frequent event or a singular one. If the seat removal happened multiple times, but you like the sound of "we removed" better than the sound of "we would remove," you can easily accommodate your preference by adding a simple clause (such as "each night") to the sentence to indicate its recurring nature:
Each night, after cooking our dinner, we removed the middle seat where Chris, Tim, Ray and Eli could sleep.
The prefatory phrase "each night" conveys the intended sense of repetition at least as effectively as "we would remove" does, so readers will have no reason to erroneously suppose that you are describing something that happened only once.