Actually, to stay on topic, since the question was how to best use non-chicken eggs, I'd say duck and goose eggs are best fried or used in omelette - haven't tried them for baking - not as good as chicken eggs when boiled. Quail eggs are easier to handle boiled than to crack for frying, baking etc.
For most people chicken eggs will be more familiar and more easily available, so in a way maybe the best use of non-chicken eggs in general would be in applications where you will notice and appreciate their difference. For instance, what is the point in cracking 15 quail eggs for a cake where you could use three chicken eggs? Garnishing a salad with boiled quail eggs, or having lots of tiny fried eggs on toast for instance, will be more enjoyable because then you can tell they are not chicken eggs. Similarly, I like frying duck or goose eggs because then the size difference is obvious, and any subtle differences in taste and texture are more noticeable.
Of course, if local availability means any of the non-chicken eggs are your standard or default eggs, that would make it different.
By the way, gull eggs used to be collected in certain parts of Norway (presumably other places as well), they were considered particularly good soft boiled just by themselves or for garnishing soups etc., and at least some people thought they made especially good waffle or pancake batter. It's been a long time since I had any (gull populations are not doing well, I'm not sure egg harvesting is allowed any more), but to be honest I'm not sure I could tell any difference between gull and chicken eggs. Apart from the beautiful shells..