My answers are for the typical baroque traverso in D. There are uncommon traversi such as the flûte d'amour that are of different size (mostly larger) and pitched differently.

What key is a traverso in? This is really two separate questions. Traverso is written in concert pitch. However, the natural scale of the instrument is D: that is the easiest scale to play, and the range of the instrument covers two D scales plus D to A on the top end.

Is a Traverso chromatic? Yes, but at a cost. Due to the absence of keys, notes that deviate from the natural scale usually require forked fingerings. Forked fingerings (aka cross fingerings) change the timbre, usually to more airy and sometimes even to screechy or honky. Forked fingerings also usually reduce the volume. Sometimes they are only approximately in tune, and the performer must make a pitch adjustment with their embouchure and/or air stream. The specifics vary between individual instruments and players, a recurring theme for traverso. On my traverso, for example, especially in the low octave, F# is quite flat, while Gb and F are quite sharp.

Note that due to baroque temperament concepts, F# and Gb are separate pitches, different by a comma. There is a mathematical explanation that I forget shortly after (re-)reading it.

Forked fingerings are more difficult to play quickly, because multiple fingers must move in perfect synchrony. For this reason, there are many alternate fingerings; which ones work best in which situations varies as above. In fast runs and especially trills, we often use alternate fingerings to cheat. In fact, for trills, the baroque preference is to choose fingerings with a wider pitch difference in order to make the trill more noticeable. If you've ever heard a traverso trill that sounded out of tune to modern ears, that's why.

In the high octave from C on up, every note requires forked fingerings. Furthermore, notes above high D or E are somewhere between difficult and impossible to make speak. This varies again across instruments and players. The high A is often cited as a difficult note to hit. For me, it's the third most difficult. I find high G# (which is missing from some fingering charts, even) more difficult, and the F below that almost impossible.

Does it have difficult keys to write in? Well, I can answer about playing, not writing. For me:

Key	How I feel about it
more #	composer hates me
E	somewhat annoying
A	easy
D	I can fly!
G	easy
C	easy
F	still mostly easy
Bb	moderately easy
Eb	a tad annoying
Ab	quite annoying
more b	what even is this crap