I do recommend avoiding quests that you find boring, because games are supposed to be fun, not boring. But if you want to know how I solve these quests, I'll lay out my strategy.
For the bird thief quest:
The way I do it: the NPC tells me to go, for example, 4 km, in some direction, say NorthWest. This means 31-40 tiles so I walk 31 squares in that direction, mark a wilderness map tile as being 31, then go further to mark tile 40. I compare this ten by ten region with the F6 map: you're guaranteed to be within the circle ont he F6 map but you're not guaranteed to be in this 10x10 square, because it might just as well be 25 tiles exactly northwest and the another 6-16 west of that, but if you start at the 31 point and then imagine a line perpendicular to that , that will tell you the limit of where it can be, and a similar perpendicular line in the next direction.
So now you have a parallelogram and a circle and you only have to search the tiles where they intersect -- they might not even intersect that much. You only have to look at the tiles that match the description like "coniferous forest bordering on heathland to the east" which doesn't mean that the traps will be right near the border between the heathland and the coniferous forest, it just means "somewhere in a coniferous forest tile which has a heathland tile east of it".
If the tile is spruce mire or coniferous forest, then your work is harder because you won't be able to just zoom in and immediately spot it; you'll need to climb a tree. drop all of your equipment and make sure you are unburdened and clime up. You can use the mouse wheel to scroll out. The loop snares are a little hard to spot but the feathers show up very well.
If I don't see any, I zoom out and drop a branch on the wilderness map so I will know that I have searched the tile already. I prefer this to cluttering up the F6 map with markers.
The lost pet quest is harder, and it's not worth it for a sheep, you only get 5 squirrel hides for a sheep, but for a cow or a bull or a reindeer it is worth it - you can get about 17 squirrel hides for a cow. I do the parallelogram circle intersection thing again and then this is kind of tedious but I watch youtube videos while I'm doing it so it's less tedious, but I go into stealth mode to train stealth and I just walk in a straight line from one edge to the other, zooming out every 64 steps to drop a branch so that later i'll be able to see which tiles i've been through and then I literally just explore the whole thing, systematically and methodically. Combing through a series of zoomed in maps like this spawns a lot of game so I usually wind up with some kills as a result of all this, and it trains my stealth and tracking up by a lot too.
I will try to pictures of the circle parallelogram thing next time I get the quest