Answer 1 - they're different -sax, equal temperament, like a guitar, but trumpet needs adjusting by the player, as the natural notes are from the harmonic series so are 'slightly out of tune' (desafinado?) and yes, a 'dirty' note can be produced by both. I hope you mean a note 'in the cracks', rather like a bent note on guitar.
Answer 2 - on sax, once you've found your embouchure, it varies little for each basic note, which is found using the keys. On trumpet, the lips do much of the work, tightening as the notes get higher. Once you're on a particular overtone, the valves will lower it by a semitone, a tone, etc.
Suggestion - as ever, and as David says, a teacher will sort out the conundrums far more easily than any other method. You can ask and get the answer there and then, with demonstrations and explanations. Even a couple of lessons are worth their weight in gold.
As Matt says, sax is going to be the better option, as it can be picked up at will without the previous hard work having to be done again - the embouchure for trumpet seems to be more difficult to keep without using daily. The other thing is that on sax, there's a good two octaves available without too much effort; on trumpet, as you found, two octaves is hard work for the lips.