All instruments are equally difficult for different reasons. You can't escape by choosing an "easy" instrument because there are none.
All pitches from all instruments have overtones. It is impossible not to use an overtone as they are inherent in the physical properties of vibration that actuate the pitch. It is precisely the reliance on the ear to negotiate the harmonic series on brass instruments that make them so difficult to play. I believe you're looking for the term "partial" - as in a particular note from the harmonic series.
Brass players have a saying that if you can hear it, you can play it. Next time you go to play, hear the Bb first and then sing the pitch, making sure it doesn't waver. Next, bring the mouthpiece (not the whole trumpet!) up to your mouth and buzz the Bb into just the mouthpiece. Do this several times until the pitch does not waver. Next, use your lips to bend away (higher and lower) from the pitch and see if you can come exactly back to it (it's okay to use the computer as a drone if necessary.) Repeat this a few times. Now add the mouthpiece to your trumpet and play.
You may want to read my answer on "Audiation" as I think that might help a little bit - at least in terms of understanding why what I just told you is important.
Like the trumpet, the saxophone is a transposing instrument, so it's important to specify if you're looking for written range or sounding range. For all saxophones, the written range is Bb3 - G6. As alluded to earlier in my answer, it is impossible not to use overtones when playing the saxophone.
Yes, the saxophone is just as difficult to play in tune as all other instruments. You are using different muscles in a different embouchure and that requires different technique.
On brass instruments, altering the pitch without changing the fingering is actually a lip-exercise used to develop flexibility and intonation. First, do not worry that you're having a hard time with the first note - almost everyone feels lost at first when beginning to learn a brass instrument. Most students have a much easier time playing the "G" (written) because it is easier to slot at first. Because the trumpet mouthpiece is small, the low range is actually difficult to produce satisfactory sounds and takes easing into.
To answer your question more directly, generally you can bend a full 1/2 step (semitone) in either direction. In the lower range, and especially on larger brass instruments, you can bend as much as a perfect fifth (as is the case of the tuba.)
I hope that answers your questions.
As a side note, I think you made a good move going for brass - now everyone doesn't have to hear you complain about reeds. :P
Hope that helps!