The D in 1st position - it varies from horn to horn but more often than not yes it'll be a little flat. If you're playing the D as the third of a Bb major chord, then you're already adjusted, easy. If you need a really in-tune D, either 1) tune the whole horn such that 1st position is not quite "all the way in" so you have some room to sharpen the D, 2) use the D in 4th position instead.

Lowering the thirds of chords when you're playing them is generally not something people worry about until they're serious players. And it's really more of an ear training thing than a neuroticism thing. The exercise is to play a static drone over some speakers (say a D), and then play each note of a D major chord up the range, sliding in an out until you can sort of feel the overtones locking in. On the F# you'll feel the lock-in at a flatter position that F# normally is. And the idea is that this proprioceptive sense of intonation will then carry over to your playing.