A few answers, Lithuania has next to no birding tradition - in the entire country, there are barely a handful of active birders. My local patch, which I often post about, is usually totally devoid of other birders, one Finnish guy also covers it and very occasionally you might get to see a Lithuanian (but that would be a mega-rarity almost!). More or less, wildlife is neither actively appreciated, but there is also very little active persecution. Wanton destruction would be very rare.
Commonest garden bird? Depends which garden and season - Tree Sparrow and Great Tit take the honours in winter, though get up to five woodpecker species around my feeders too (though not strictly in a garden). In summer, Pied Flycatchers are perhaps the most common, occupying boxes very close together.
Commonest bird of prey (by regularity of sightings) - local patch, it would be first Marsh Harrier, then Buzzard, then White-tailed Eagle. On my land, first Marsh Harrier or Lesser Spotted Eagle, then Buzzard.