Changing key signature to only one instrument (staff)

To add a key signature for one instrument only, hold Ctrl while adding the key (FWIW, this is in the chapter on key signatures, the last paragraph of the section "Initial Key signature"). Since this presumably isn't really mean to be transposing for horns, be sure to use the "open" or "atonal" key signature (the one with the greyed-out "X"), not the regular "C major" one.

A trombone having no key signature in a symphony with a key signature is most unusual. I haven't seen it before.

Since Shostakovitch's 10th is still under copyright, I haven't seen the score before so there are a few I would look for to make sure the no Key Signature trombone is correct. First, check the translation of the instrument you call a trombone. If the untranslated name of the instrument does not have the word trombone in it, it is most likely another instrument. For example, Tromba is a Trumpet. If the clef is a treble clef it is extremely unlikely you have a trombone part you are looking at. Trombones in symphonies are always in C or Do, if a key is indicated, it is not a trombone. Look at later movements to see if there is a different instrumentation and see if the trombone is shown without a key there also, while other C instruments have a key.